Datasets:

nanatata

/

hja3

like 0

3stpes_51

testmini

Understanding Length (Using a Ruler)

As shown in the figure, use a ruler to measure the length of the rectangle's side. What is the length of BC in cm?

A. 3; B. 4; C. 5; D. No correct answer

A

3steps_1

Understanding Length (Using a Ruler): 1. Length is usually measured using a ruler. 2. When using a ruler for measurement, first determine the measurement unit of the ruler. If there is no special indication, the default unit is centimeters. 3. If one end of the line segment being measured is aligned with the 0 mark on the ruler, the length of the line segment is the direct reading from the other end. 4. If one end of the line segment being measured is not aligned with the 0 mark, the length of the line segment is the difference between the readings at both ends.

3stpes_51

testmini

Area of Rectangles

As shown in the figure, the area of rectangle ABCD is 6 cm². What is the length of CD?

A. 2; B. 3; C. 4; D. 5; E. No correct answer

A

3steps_2

Area of Rectangles: 1. The area of a rectangle is equal to its length × width, expressed as: S = ab.

3stpes_51

testmini

Representing Positions Using Ordered Pairs

As shown in the figure, point B can be represented by the coordinates (2, 2). What are the coordinates of point D?

A. (4, 5); B. (5, 4); C. (2, 2); D. No correct answer

B

3steps_3

Representing Positions Using Ordered Pairs: 1. Definition of an Ordered Pair: An ordered pair is a combination of two numbers, typically used to represent the position of a point in a Cartesian coordinate system. The two numbers in the ordered pair represent values on different directions or axes. 2. Composition of an Ordered Pair: In an ordered pair (x, y), the first number x represents the column number, which is the horizontal position; the second number y represents the row number, which is the vertical position. This notation follows the "horizontal first, then vertical" convention. 3. Notation of an Ordered Pair: When using an ordered pair to represent a point's position, the numbers or letters representing the column and row need to be enclosed in parentheses, for example, (2, 5) represents the position at column 2, row 5. 4. Uniqueness of an Ordered Pair: Each ordered pair uniquely determines a position, and conversely, a position can be uniquely represented by an ordered pair.

3stpes_51

testmini

Understanding Length (Using a Ruler);Area of Rectangles;Representing Positions Using Ordered Pairs

As shown in the figure, point B can be represented by the coordinates (2, 2), and the area of the rectangle is 6 cm². What are the coordinates of point D?

A. (5, 4); B. (5, 3); C. (4, 5); D. No correct answer

A

3steps_multi

Understanding Length (Using a Ruler): 1. Length is usually measured using a ruler. 2. When using a ruler for measurement, first determine the measurement unit of the ruler. If there is no special indication, the default unit is centimeters. 3. If one end of the line segment being measured is aligned with the 0 mark on the ruler, the length of the line segment is the direct reading from the other end. 4. If one end of the line segment being measured is not aligned with the 0 mark, the length of the line segment is the difference between the readings at both ends. Area of Rectangles: 1. The area of a rectangle is equal to its length × width, expressed as: S = ab. Representing Positions Using Ordered Pairs: 1. Definition of an Ordered Pair: An ordered pair is a combination of two numbers, typically used to represent the position of a point in a Cartesian coordinate system. The two numbers in the ordered pair represent values on different directions or axes. 2. Composition of an Ordered Pair: In an ordered pair (x, y), the first number x represents the column number, which is the horizontal position; the second number y represents the row number, which is the vertical position. This notation follows the "horizontal first, then vertical" convention. 3. Notation of an Ordered Pair: When using an ordered pair to represent a point's position, the numbers or letters representing the column and row need to be enclosed in parentheses, for example, (2, 5) represents the position at column 2, row 5. 4. Uniqueness of an Ordered Pair: Each ordered pair uniquely determines a position, and conversely, a position can be uniquely represented by an ordered pair.

3stpes_52

testmini

Observing Objects

Lisa arranges several 1 cm edge-length cubic blocks into a rectangular cuboid. As shown in the figure, the images are views from the front and the top of the rectangular cuboid. What are the length, width, and height of this rectangular cuboid respectively in cm?

A. 4, 5, 6; B. 5, 4, 3; C. 4, 3, 2; D. No correct answer

C

3steps_1

Observing Objects: 1. When observing an object from different angles, you will see two or three adjacent faces, and the shape observed on each face is different. 2. The top, front, and side faces of a rectangular cubiod or cube are defined by convention. The face facing upward is called the top face, the face facing the observer is called the front face, and the two side faces are called the side faces. It is impossible to see opposite faces of a rectangular cubiod or cube at the same time. 3. When observing a combination of objects, not only should the shape of each object be considered, but also the positional relationships between the objects. If an object is blocked by another object, the blocked part will be difficult to observe. 4. No matter which face a cube or sphere is observed from, the shape seen is always the same.

3stpes_52

testmini

Volume and Capacity of Rectangular Cuboids

As shown in the figure, Lisa uses several 1 cm edge length cubic wooden blocks to form the following rectangular cuboid. What is its volume in cm³?

A. 10; B. 18; C. 24; D. No correct answer

C

3steps_2

Volume and Capacity of Rectangular Cuboids: 1. The formula for calculating the volume of a rectangular cubiod: The volume of a rectangular cubed = length × width × height. 2. In terms of letters: If V represents the volume of the rectangular cubiod, and a, b, h represent the length, width, and height of the rectangular cubed respectively, then the formula for the volume of the rectangular cubed can be written as: V = abh. 3. The volume of a rectangular cubiod is equal to the base area multiplied by the height. 4. The capacity of a rectangular cubiod usually refers to the amount of space a rectangular container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_52

testmini

Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters)

What is the volume of the rectangular cuboid shown in the diagram, in cubic decimeters (dm³)?

A. 0.024; B. 0.012; C. 0.02; D. 0.027; E. No correct answer

A

3steps_3

Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters): 1. Conversion rates between volume units: 1 cubic meter (m³) = 1000 cubic decimeters (dm³); 1 cubic decimeter (dm³) = 1000 cubic centimeters (cm³); 1 cubic decimeter (dm³) = 1 liter (L); 1 cubic centimeter (cm³) = 1 milliliter (mL) 2. Converting between Different Units: To convert from a larger unit to a smaller unit, multiply by the conversion rate. To convert from a smaller unit to a larger unit, divide by the conversion rate. 3. To convert cubic meters to cubic decimeters: multiply by 1000. 4. To convert cubic decimeters to cubic centimeters: multiply by 1000. 5. To convert cubic decimeters to liters: 1 cubic decimeter equals 1 liter. 6. To convert cubic centimeters to milliliters: 1 cubic centimeter equals 1 milliliter. 7. To convert milliliters to cubic centimeters: 1 milliliter equals 1 cubic centimeter. 8. To convert liters to cubic decimeters: 1 liter equals 1 cubic decimeter. 9. To convert cubic decimeters to cubic meters: divide by 1000. 10. To convert cubic centimeters to cubic decimeters: divide by 1000. 11. To convert cubic decimeters to cubic meters: divide by 1000.

3stpes_52

testmini

Observing Objects;Volume and Capacity of Rectangular Cuboids;Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters)

Lisa used several 1 cm³ cubic wooden blocks to form a rectangular cuboid. As shown in the figure, the images are views from the front and the top of the rectangular cuboid. What is the volume of this rectangular cuboid in cubic decimeters (dm³)?

A. 0.024; B. 0.012; C. 0.02; D. 0.027; E. No correct answer

A

3steps_multi

Observing Objects: 1. When observing an object from different angles, you will see two or three adjacent faces, and the shape observed on each face is different. 2. The top, front, and side faces of a rectangular cubiod or cube are defined by convention. The face facing upward is called the top face, the face facing the observer is called the front face, and the two side faces are called the side faces. It is impossible to see opposite faces of a rectangular cubiod or cube at the same time. 3. When observing a combination of objects, not only should the shape of each object be considered, but also the positional relationships between the objects. If an object is blocked by another object, the blocked part will be difficult to observe. 4. No matter which face a cube or sphere is observed from, the shape seen is always the same. Volume and Capacity of Rectangular Cuboids: 1. The formula for calculating the volume of a rectangular cubiod: The volume of a rectangular cubed = length × width × height. 2. In terms of letters: If V represents the volume of the rectangular cubiod, and a, b, h represent the length, width, and height of the rectangular cubed respectively, then the formula for the volume of the rectangular cubed can be written as: V = abh. 3. The volume of a rectangular cubiod is equal to the base area multiplied by the height. 4. The capacity of a rectangular cubiod usually refers to the amount of space a rectangular container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods. Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters): 1. Conversion rates between volume units: 1 cubic meter (m³) = 1000 cubic decimeters (dm³); 1 cubic decimeter (dm³) = 1000 cubic centimeters (cm³); 1 cubic decimeter (dm³) = 1 liter (L); 1 cubic centimeter (cm³) = 1 milliliter (mL) 2. Converting between Different Units: To convert from a larger unit to a smaller unit, multiply by the conversion rate. To convert from a smaller unit to a larger unit, divide by the conversion rate. 3. To convert cubic meters to cubic decimeters: multiply by 1000. 4. To convert cubic decimeters to cubic centimeters: multiply by 1000. 5. To convert cubic decimeters to liters: 1 cubic decimeter equals 1 liter. 6. To convert cubic centimeters to milliliters: 1 cubic centimeter equals 1 milliliter. 7. To convert milliliters to cubic centimeters: 1 milliliter equals 1 cubic centimeter. 8. To convert liters to cubic decimeters: 1 liter equals 1 cubic decimeter. 9. To convert cubic decimeters to cubic meters: divide by 1000. 10. To convert cubic centimeters to cubic decimeters: divide by 1000. 11. To convert cubic decimeters to cubic meters: divide by 1000.

3stpes_53

testmini

Combining and Dividing Solids

As shown in the figure, a cylindrical piece of wood is cut vertically along the diameter of its base into two parts. Compared to the original cylinder, the surface area increases by 60 dm². What is the area of the rectangle ABCD in the figure in dm²?

A. 30; B. 40; C. 50; D. 60; E. No correct answer

A

3steps_1

Combining and Dividing Solids: 1. When two or more geometric solids are combined to form a new solid, the volume of the new solid equals the sum of the volumes of the original solids. However, the surface area of the new solid is less than the sum of the surface areas of the original solids. If the overlapping area is S, the decrease in surface area is 2S. 2. When a geometric solid is cut into parts, the sum of the volumes of the parts equals the volume of the original solid. However, the sum of the surface areas of the parts is greater than the surface area of the original solid. If the cutting area is S, the increase in surface area is 2S.

3stpes_53

testmini

Area of Rectangles

A cylindrical wooden log is cut vertically along the diameter of its base into two parts (as shown in the figure), resulting in two rectangular cut surfaces. What is the diameter of the base of this cylindrical wooden log in decimeters (dm)?

A. 6; B. 4; C. 5; D. 3; E. No correct answer

A

3steps_2

Area of Rectangles: 1. The area of a rectangle is equal to its length × width, expressed as: S = ab.

3stpes_53

testmini

Surface Area of Cylinders

A cylindrical wooden log is cut vertically along the diameter of its base into two parts (as shown in the figure), resulting in two rectangular cut surfaces. What was the original surface area of the cylindrical wooden log in square decimeters (dm²)?

A. 18π; B. 48π; C. 72π; D. 132π; E. No correct answer

B

3steps_3

Surface Area of Cylinders: 1. The lateral surface area of a cylinder = circumference of the base × height. 2. The area of the base of a cylinder = π × radius squared. 3. The surface area of a cylinder refers to the sum of its lateral surface area and the areas of its two bases.

3stpes_53

testmini

Combining and Dividing Solids;Area of Rectangles;Surface Area of Cylinders

A cylindrical wooden log with a height of 5 dm is cut vertically along the diameter of its base into two parts. Compared to the original cylinder, the surface area increases by 60 dm². What was the original surface area of the cylindrical log in dm²?

A. 18π; B. 48π; C. 72π; D. 132π; E. No correct answer

B

3steps_multi

Combining and Dividing Solids: 1. When two or more geometric solids are combined to form a new solid, the volume of the new solid equals the sum of the volumes of the original solids. However, the surface area of the new solid is less than the sum of the surface areas of the original solids. If the overlapping area is S, the decrease in surface area is 2S. 2. When a geometric solid is cut into parts, the sum of the volumes of the parts equals the volume of the original solid. However, the sum of the surface areas of the parts is greater than the surface area of the original solid. If the cutting area is S, the increase in surface area is 2S. Area of Rectangles: 1. The area of a rectangle is equal to its length × width, expressed as: S = ab. Surface Area of Cylinders: 1. The lateral surface area of a cylinder = circumference of the base × height. 2. The area of the base of a cylinder = π × radius squared. 3. The surface area of a cylinder refers to the sum of its lateral surface area and the areas of its two bases.

3stpes_54

testmini

Perimeter of Rectangles

As shown in the figure, the area of the rectangle is 630 cm². What is the width of the rectangle in cm?

A. 21 cm; B. 30 cm; C. 25 cm; D. No correct answer

A

3steps_1

Perimeter of Rectangles: 1. A rectangle has equal opposite sides, and the perimeter of a rectangle = (length + width) × 2 (C = 2(a+b)).

3stpes_54

testmini

Division of Plane Figures

From the rectangle shown in the figure, a largest possible square is cut out. What is the side length of the square ( ) cm?

A. 16; B. 21; C. 32; D. No correct answer

B

3steps_2

Division of Plane Figures: 1. Dividing a geometric figure into several parts according to certain requirements is called the partitioning of the figure. 2. When a plane figure is divided into two or more new geometric figures, the area of the original figure equals the sum of the areas of the new figures, but the perimeter of the original figure will be shorter than the sum of the perimeters of the resulting figures. If the total length of the dividing lines is a, then the increase in the perimeter after division is 2a.

3stpes_54

testmini

Area of Squares

From the rectangle shown in the figure, a largest possible square is cut out. What is the area of the square in cm²?

A. 410; B. 198; C. 441; D. No correct answer

C

3steps_3

Area of Squares: 1. The area of a square is equal to the square of its side length: S = a * a. 2. The area of a square is equal to the square of the length of its diagonal divided by two.

3stpes_54

testmini

Perimeter of Rectangles;Division of Plane Figures;Area of Squares

As shown in the figure, the area of the rectangular piece of paper is 630 cm². If the largest possible square is cut from this piece of paper, what is the area of this square in square centimeters?

A. 30; B. 70; C. 441; D. 900; E. No correct answer

C

3steps_multi

Perimeter of Rectangles: 1. A rectangle has equal opposite sides, and the perimeter of a rectangle = (length + width) × 2 (C = 2(a+b)). Division of Plane Figures: 1. Dividing a geometric figure into several parts according to certain requirements is called the partitioning of the figure. 2. When a plane figure is divided into two or more new geometric figures, the area of the original figure equals the sum of the areas of the new figures, but the perimeter of the original figure will be shorter than the sum of the perimeters of the resulting figures. If the total length of the dividing lines is a, then the increase in the perimeter after division is 2a. Area of Squares: 1. The area of a square is equal to the square of its side length: S = a * a. 2. The area of a square is equal to the square of the length of its diagonal divided by two.

3stpes_55

testmini

Understanding Angles (Using a Protractor)

As shown in the figure, using a protractor to measure one of the angles of the triangular glass fragment, the measured angle is ()°?

A. 40; B. 50; C. 60; D. No correct answer

A

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_55

testmini

Sum of Interior Angles of Triangles

As shown in the figure, a piece of triangular glass is missing. What is the missing angle of this triangle ( )°?

A. 105; B. 95; C. 120; D. No correct answer

A

3steps_2

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_55

testmini

Properties and Understanding of Triangles

The three angles of a triangle are shown in the figure. What type of triangle is this?

A. Right-angled; B. Acute-angled; C. Obtuse-angled; D. No correct answer

C

3steps_3

Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_55

testmini

Understanding Angles (Using a Protractor);Sum of Interior Angles of Triangles;Properties and Understanding of Triangles

A piece of triangular glass, with fragments shown in the figure, originally was a () triangle.

A. Obtuse; B. Acute; C. Right; D. Cannot be determined; E. No correct answer

A

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_56

testmini

Understanding Triangular Rulers

As shown in the figure, using a triangular ruler to measure the angle ∠A, what is the degree of ∠A?

A. 30°; B. 95°; C. 110°; D. No correct answer

A

3steps_1

Understanding Triangular Rulers: 1. A triangular ruler is a triangular-shaped tool commonly used for measuring angles, drawing straight lines, and performing geometric constructions. 2. There are two types of triangular ruler: Isosceles right trianglular ruler: One angle is 90°, and the other two angles are 45° and 45°. Scalene right triangular ruler: One angle is 90°, and the other two angles are 30° and 60°.

3stpes_56

testmini

Sum of Interior Angles of Triangles

A piece of triangular glass (as shown in the figure) has a missing angle of ()°.

A. 105; B. 95; C. 110; D. No correct answer

C

3steps_2

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_56

testmini

Properties and Understanding of Triangles

The three angles of a triangle are shown in the figure. What type of triangle is this?

A. Right-angled; B. Acute-angled; C. Obtuse-angled; D. No correct answer

C

3steps_3

Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_56

testmini

Understanding Triangular Rulers;Sum of Interior Angles of Triangles;Properties and Understanding of Triangles

A piece of triangular glass, with fragments shown in the figure, originally was a () triangle.

A. Acute; B. Right; C. Obtuse; D. No correct answer

C

3steps_multi

Understanding Triangular Rulers: 1. A triangular ruler is a triangular-shaped tool commonly used for measuring angles, drawing straight lines, and performing geometric constructions. 2. There are two types of triangular ruler: Isosceles right trianglular ruler: One angle is 90°, and the other two angles are 45° and 45°. Scalene right triangular ruler: One angle is 90°, and the other two angles are 30° and 60°. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_57

testmini

Understanding Angles (Using a Protractor)

As shown in the figure, use a protractor to measure one of the angles of the triangular glass shard. The measured angle is ()°?

A. 40; B. 50; C. 60; D. No correct answer

A

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_57

testmini

Sum of Interior Angles of Triangles

As shown in the figure, a piece of triangular glass is missing. What is the measure of the missing angle in this triangle ( )°?

A. 105; B. 90; C. 110; D. No correct answer

B

3steps_2

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_57

testmini

Properties and Understanding of Triangles

The three angles of a triangle are shown in the figure. What type of triangle is this?

A. Right-angled; B. Acute-angled; C. Obtuse-angled; D. No correct answer

A

3steps_3

Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_57

testmini

Understanding Angles (Using a Protractor);Sum of Interior Angles of Triangles;Properties and Understanding of Triangles

As shown in the figure, a piece of triangular glass has shattered, leaving behind fragments. What was its original shape?

A. Acute triangle; B. Right triangle; C. Obtuse triangle; D. No correct answer

B

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_58

testmini

Understanding Length (Using a Ruler)

As shown in the figure, using a ruler to measure the side lengths of triangle ABC, what type of triangle is triangle ABC?

A. Isosceles; B. Right-angled; C. Equilateral; D. No correct answer

A

3steps_1

Understanding Length (Using a Ruler): 1. Length is usually measured using a ruler. 2. When using a ruler for measurement, first determine the measurement unit of the ruler. If there is no special indication, the default unit is centimeters. 3. If one end of the line segment being measured is aligned with the 0 mark on the ruler, the length of the line segment is the direct reading from the other end. 4. If one end of the line segment being measured is not aligned with the 0 mark, the length of the line segment is the difference between the readings at both ends.

3stpes_58

testmini

Sum of Interior Angles of Triangles

As shown in the figure, triangle ABC is an isosceles triangle. What is the measure of ∠B in degrees?

A. 25; B. 35; C. 10; D. No correct answer

A

3steps_2

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_58

testmini

Calculation and Comparison of Angles

As shown in the figure, triangle ABC is an isosceles triangle, ∠B = 25°, then the angle of ∠1 is ( )

A. 95°; B. 145°; C. 120°; D. No correct answer

C

3steps_3

Calculation and Comparison of Angles: 1. Sum and difference of angles: Adding multiple angles gives the total angle, and subtracting gives the difference angle. 2. The size of an angle is not related to the length of its two sides, but only to the size of the angle's opening. 3. Comparison of angles: The size of angles can be compared directly by their degrees. Straight angle > obtuse angle > right angle > acute angle.

3stpes_58

testmini

Understanding Length (Using a Ruler);Sum of Interior Angles of Triangles;Calculation and Comparison of Angles

As shown in the figure, using a ruler to measure the side length AB of triangle ABC, from the information in the figure, it can be calculated that ∠1 is ( ).

A. 95°; B. 145°; C. 120°; D. No correct answer

C

3steps_multi

Understanding Length (Using a Ruler): 1. Length is usually measured using a ruler. 2. When using a ruler for measurement, first determine the measurement unit of the ruler. If there is no special indication, the default unit is centimeters. 3. If one end of the line segment being measured is aligned with the 0 mark on the ruler, the length of the line segment is the direct reading from the other end. 4. If one end of the line segment being measured is not aligned with the 0 mark, the length of the line segment is the difference between the readings at both ends. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Calculation and Comparison of Angles: 1. Sum and difference of angles: Adding multiple angles gives the total angle, and subtracting gives the difference angle. 2. The size of an angle is not related to the length of its two sides, but only to the size of the angle's opening. 3. Comparison of angles: The size of angles can be compared directly by their degrees. Straight angle > obtuse angle > right angle > acute angle.

3stpes_59

testmini

Understanding Angles (Using a Protractor)

As shown in the figure, the protractor measures the degree of ∠B as ()°?

A. 42; B. 50; C. 60; D. No correct answer

A

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_59

testmini

Properties and Understanding of Triangles

As shown in the figure, in the right triangle ABC, ∠A = ( )°

A. 42; B. 90; C. 48; D. No correct answer

B

3steps_2

Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_59

testmini

Sum of Interior Angles of Triangles

As shown in the figure, in the right triangle CAB, ∠A = 90°, then ∠C = ( )°

A. 48°; B. 58°; C. 45°; D. 68°; E. No correct answer

A

3steps_3

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_59

testmini

Understanding Angles (Using a Protractor);Properties and Understanding of Triangles;Sum of Interior Angles of Triangles

As shown in the figure, in the right triangle CAB, ∠C = ( )°.

A. 48; B. 58; C. 45; D. 68; E. No correct answer

A

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_60

testmini

Understanding Angles (Using a Protractor)

As shown in the figure, what is the degree measure of the angle measured with the protractor? ( )°?

A. 110; B. 100; C. 90; D. No correct answer

A

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_60

testmini

Calculation and Comparison of Angles

As shown in the figure, ∠2 = ()°

A. 52; B. 70; C. 72; D. 38; E. No correct answer

B

3steps_2

Calculation and Comparison of Angles: 1. Sum and difference of angles: Adding multiple angles gives the total angle, and subtracting gives the difference angle. 2. The size of an angle is not related to the length of its two sides, but only to the size of the angle's opening. 3. Comparison of angles: The size of angles can be compared directly by their degrees. Straight angle > obtuse angle > right angle > acute angle.

3stpes_60

testmini

Sum of Interior Angles of Triangles

As shown in the figure, it is known that ∠2 = 70°. Then ∠1 = ( )°

A. 52; B. 70; C. 72; D. 38; E. No correct answer

C

3steps_3

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_60

testmini

Understanding Angles (Using a Protractor);Calculation and Comparison of Angles;Sum of Interior Angles of Triangles

As shown in the figure, use a protractor to measure the angle in the diagram and calculate ∠1 = ()°.

A. 52; B. 70; C. 72; D. 38; E. No correct answer

C

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Calculation and Comparison of Angles: 1. Sum and difference of angles: Adding multiple angles gives the total angle, and subtracting gives the difference angle. 2. The size of an angle is not related to the length of its two sides, but only to the size of the angle's opening. 3. Comparison of angles: The size of angles can be compared directly by their degrees. Straight angle > obtuse angle > right angle > acute angle. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_61

testmini

Understanding Angles (Using a Protractor)

As shown in the figure, what is the degree measure of ∠3 using the protractor? ( )°?

A. 40; B. 50; C. 60; D. No correct answer

B

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_61

testmini

Sum of Interior Angles of Triangles

As shown in the figure, in the triangle, ∠2 = ( )

A. 95°; B. 85°; C. 105°; D. No correct answer

B

3steps_2

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_61

testmini

Calculation and Comparison of Angles

Calculate the degree of angle ∠1 based on the degrees of the other angles shown below ().

A. 95°; B. 85°; C. 105°; D. No correct answer

A

3steps_3

Calculation and Comparison of Angles: 1. Sum and difference of angles: Adding multiple angles gives the total angle, and subtracting gives the difference angle. 2. The size of an angle is not related to the length of its two sides, but only to the size of the angle's opening. 3. Comparison of angles: The size of angles can be compared directly by their degrees. Straight angle > obtuse angle > right angle > acute angle.

3stpes_61

testmini

Understanding Angles (Using a Protractor);Sum of Interior Angles of Triangles;Calculation and Comparison of Angles

As shown in the figure, use a protractor to measure the angle in the diagram. Given that ∠4 = 45°, calculate ∠1 = ( ).

A. 95°; B. 85°; C. 105°; D. No correct answer

A

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Calculation and Comparison of Angles: 1. Sum and difference of angles: Adding multiple angles gives the total angle, and subtracting gives the difference angle. 2. The size of an angle is not related to the length of its two sides, but only to the size of the angle's opening. 3. Comparison of angles: The size of angles can be compared directly by their degrees. Straight angle > obtuse angle > right angle > acute angle.

3stpes_62

testmini

Understanding Angles (Using a Protractor)

As shown in the figure, what is the measure of ∠B using the protractor? ( )°?

A. 40; B. 50; C. 60; D. No correct answer

B

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_62

testmini

Sum of Interior Angles of Triangles

As shown in the figure, in △ABC, what is ∠A + ∠C?

A. 150°; B. 140°; C. 130°; D. No correct answer

C

3steps_2

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_62

testmini

Sum of Interior Angles of Other Polygons

As shown in the figure, in △ABC, ∠A + ∠C = 130°. If ∠B is cut along the dashed line in the figure, then ∠1 + ∠2 equals ( )

A. 130°; B. 230°; C. 270°; D. 310°; E. No correct answer

B

3steps_3

Sum of Interior Angles of Other Polygons: 1. The sum of the interior angles of a quadrilateral is 360°, and the sum of the interior angles of a pentagon is 540°. 2. The sum of the interior angles of a polygon = 180° × (number of sides - 2). 3. The sum of the interior angles of an n-sided polygon = (n - 2) × 180°. 4. A polygon with all sides of equal length and all interior angles equal is called a regular polygon.

3stpes_62

testmini

Understanding Angles (Using a Protractor);Sum of Interior Angles of Triangles;Sum of Interior Angles of Other Polygons

As shown in the figure, use a protractor to measure the degree of ∠B in △ABC. If ∠B is cut along the dashed line in the figure, then ∠1 + ∠2 equals ( ).

A. 130°; B. 230°; C. 270°; D. 310°; E. No correct answer

B

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Sum of Interior Angles of Other Polygons: 1. The sum of the interior angles of a quadrilateral is 360°, and the sum of the interior angles of a pentagon is 540°. 2. The sum of the interior angles of a polygon = 180° × (number of sides - 2). 3. The sum of the interior angles of an n-sided polygon = (n - 2) × 180°. 4. A polygon with all sides of equal length and all interior angles equal is called a regular polygon.

3stpes_63

testmini

Understanding Angles (Using a Protractor)

As shown in the figure, what is the measure of the angle using the protractor? ( )°?

A. 67; B. 50; C. 60; D. No correct answer

A

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_63

testmini

Sum of Interior Angles of Triangles

A triangle paper has one of its corners torn off (as shown in the figure). What is the measure of the angle that was torn off? ( )

A. 50°; B. 40°; C. 67°; D. No correct answer

C

3steps_2

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_63

testmini

Properties and Understanding of Triangles

The triangle in the figure below is a ( ) triangle.

A. Equilateral; B. Isosceles; C. Obtuse; D. No correct answer

B

3steps_3

Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_63

testmini

Understanding Angles (Using a Protractor);Sum of Interior Angles of Triangles;Properties and Understanding of Triangles

A triangle paper piece has one of its corners torn off (as shown in the figure), and the degree of one of its angles is measured using a protractor. What type of triangle is this?

A. Equilateral; B. Isosceles; C. Obtuse; D. No correct answer

B

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_64

testmini

Understanding Triangular Rulers

As shown in the figure, using a triangular ruler to measure the degree of ∠3, what is the angle in degrees?

A. 30; B. 50; C. 60; D. No correct answer

A

3steps_1

Understanding Triangular Rulers: 1. A triangular ruler is a triangular-shaped tool commonly used for measuring angles, drawing straight lines, and performing geometric constructions. 2. There are two types of triangular ruler: Isosceles right trianglular ruler: One angle is 90°, and the other two angles are 45° and 45°. Scalene right triangular ruler: One angle is 90°, and the other two angles are 30° and 60°.

3stpes_64

testmini

Properties and Understanding of Triangles

In the figure, there is an isosceles triangle. Then ∠1 ( ) ∠2.

A. >; B. =; C. <; D. No correct answer

B

3steps_2

Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_64

testmini

Sum of Interior Angles of Triangles

In an isosceles triangle (as shown in the figure), ∠1 = ∠2. What is the measure of ∠1 in degrees?

A. 150°; B. 70°; C. 75°; D. No correct answer

C

3steps_3

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_64

testmini

Understanding Triangular Rulers;Properties and Understanding of Triangles;Sum of Interior Angles of Triangles

In an isosceles triangle (as shown in the figure), if the vertex angle ∠3 is measured using a triangular ruler, then ∠1 is equal to ( ) degrees.

A. 150°; B. 70°; C. 75°; D. No correct answer

C

3steps_multi

Understanding Triangular Rulers: 1. A triangular ruler is a triangular-shaped tool commonly used for measuring angles, drawing straight lines, and performing geometric constructions. 2. There are two types of triangular ruler: Isosceles right trianglular ruler: One angle is 90°, and the other two angles are 45° and 45°. Scalene right triangular ruler: One angle is 90°, and the other two angles are 30° and 60°. Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_65

testmini

Understanding Angles (Using a Protractor)

As shown in the figure, the protractor measures the degree of angle ∠B as ()°?

A. 80°; B. 60°; C. 85°; D. 70°; E. No correct answer

A

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_65

testmini

Sum of Interior Angles of Triangles

As shown in the figure, in triangle ABC, ∠A + ∠C =

A. 150°; B. 100°; C. 130°; D. No correct answer

B

3steps_2

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_65

testmini

Sum of Interior Angles of Other Polygons

As shown in the figure, in triangle ABC, it is known that ∠A + ∠C = 100°. If angle B is cut off along the line in the figure, then ∠1 + ∠2 equals ( )

A. 80°; B. 100°; C. 260°; D. 270°; E. No correct answer

C

3steps_3

Sum of Interior Angles of Other Polygons: 1. The sum of the interior angles of a quadrilateral is 360°, and the sum of the interior angles of a pentagon is 540°. 2. The sum of the interior angles of a polygon = 180° × (number of sides - 2). 3. The sum of the interior angles of an n-sided polygon = (n - 2) × 180°. 4. A polygon with all sides of equal length and all interior angles equal is called a regular polygon.

3stpes_65

testmini

Understanding Angles (Using a Protractor);Sum of Interior Angles of Triangles;Sum of Interior Angles of Other Polygons

As shown in the figure, in triangle ABC, if angle ∠B is cut off along the line in the figure, then ∠1 + ∠2 = ( ).

A. 80°; B. 100°; C. 260°; D. 270°; E. No correct answer

C

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Sum of Interior Angles of Other Polygons: 1. The sum of the interior angles of a quadrilateral is 360°, and the sum of the interior angles of a pentagon is 540°. 2. The sum of the interior angles of a polygon = 180° × (number of sides - 2). 3. The sum of the interior angles of an n-sided polygon = (n - 2) × 180°. 4. A polygon with all sides of equal length and all interior angles equal is called a regular polygon.

3stpes_66

testmini

Understanding Triangular Rulers

As shown in the figure, the measure of angle ∠1 using a triangular ruler is ( ).

A. 90°; B. 60°; C. 45°; D. 30°; E. No correct answer

B

3steps_1

Understanding Triangular Rulers: 1. A triangular ruler is a triangular-shaped tool commonly used for measuring angles, drawing straight lines, and performing geometric constructions. 2. There are two types of triangular ruler: Isosceles right trianglular ruler: One angle is 90°, and the other two angles are 45° and 45°. Scalene right triangular ruler: One angle is 90°, and the other two angles are 30° and 60°.

3stpes_66

testmini

Properties and Understanding of Triangles

The triangle in the figure is a () triangle, and ∠1 () ∠2.

A. Right, >; B. Isosceles, =; C. Obtuse, <; D. No correct answer

B

3steps_2

Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.

3stpes_66

testmini

Sum of Interior Angles of Triangles

Given that the triangle in the diagram is an isosceles triangle and ∠1 = ∠2, find the length of the third side of this triangle ( ).

A. 3cm; B. 6cm; C. 12cm; D. 8cm; E. No correct answer

B

3steps_3

Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_66

testmini

Understanding Triangular Rulers;Properties and Understanding of Triangles;Sum of Interior Angles of Triangles

As shown in the figure, use a triangular ruler to measure the angles inside the triangle in the diagram. Find the length of the third side of this triangle ( ).

A. 3cm; B. 6cm; C. 12cm; D. 8cm; E. No correct answer

B

3steps_multi

Understanding Triangular Rulers: 1. A triangular ruler is a triangular-shaped tool commonly used for measuring angles, drawing straight lines, and performing geometric constructions. 2. There are two types of triangular ruler: Isosceles right trianglular ruler: One angle is 90°, and the other two angles are 45° and 45°. Scalene right triangular ruler: One angle is 90°, and the other two angles are 30° and 60°. Properties and Understanding of Triangles: 1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence. 2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees. 3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal. 4. The sum of the interior angles of a triangle in a plane is 180 degrees. 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side. 6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side. 7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height. 8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change. Sum of Interior Angles of Triangles: 1. The sum of the interior angles of any triangle is 180°. 2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

3stpes_67

testmini

Understanding Angles (Using a Protractor)

The figure below shows a rectangular piece of paper. Using a protractor, the measure of ∠1 is ( ).

A. 50°; B. 60°; C. 65°; D. 70°; E. No correct answer

A

3steps_1

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.

3stpes_67

testmini

Folding Problems of Figures

The figure below shows a rectangular piece of paper. After folding the left side, ∠2 ( ) ∠3.

A. >; B. =; C. <; D. No correct answer

B

3steps_2

Folding Problems of Figures: 1. The folded figure exhibits the properties of a symmetrical figure, with the fold line as the axis of symmetry, and the two parts of the figure before and after folding are symmetrical about the fold line.

3stpes_67

testmini

Understanding and Representing Angles

The figure below shows a rectangular piece of paper. Using a protractor, it is measured that ∠1 = 50°. After folding the left side, ∠2 = ∠3. The measure of ∠2 is ( ).

A. 50°; B. 60°; C. 65°; D. 70°; E. No correct answer

C

3steps_3

Understanding and Representing Angles: 1. Understanding angles from a static perspective: An angle is a figure formed by two rays originating from a single point. 2. Understanding angles from a dynamic perspective: When a ray rotates around its vertex to another position, the figure formed by these two rays is called an angle. Two rays with a common endpoint form an angle, this common endpoint is called the vertex of the angle, and the two rays are called the sides of the angle. 3. Since rays extend infinitely in one direction, the length of the sides of an angle is irrelevant to the size of the angle. 4. The size of an angle can be measured and compared. 5. Straight angle: A 180° angle. When the two sides of an angle are on the same line, the angle formed is called a straight angle. Specifically, when the ray OA rotates around point O, and the terminal side is on the extension line of the initial side OA in the opposite direction, it forms a straight angle. 6. Right angle: A 90° angle. When the ray OA rotates around point O, and the terminal side is perpendicular to the initial side, it forms a right angle. Half of a straight angle is called a right angle. 7. Acute angle: An angle greater than 0° and less than 90°. An angle smaller than a right angle is called an acute angle. 8. Obtuse angle: An angle greater than 90° and less than 180°. An angle greater than a right angle and less than a straight angle is called an obtuse angle. 9. Full angle: A 360° angle. When the ray OA rotates around point O, and the terminal side coincides with the initial side, it forms a full angle.

3stpes_67

testmini

Understanding Angles (Using a Protractor);Folding Problems of Figures;Understanding and Representing Angles

The figure below shows a rectangular piece of paper. Measure the degree of ∠1 using a protractor, and then fold the left side up. What is the degree of ∠2?

A. 50°; B. 60°; C. 65°; D. 70°; E. No correct answer

C

3steps_multi

Understanding Angles (Using a Protractor): 1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale. 2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. Folding Problems of Figures: 1. The folded figure exhibits the properties of a symmetrical figure, with the fold line as the axis of symmetry, and the two parts of the figure before and after folding are symmetrical about the fold line. Understanding and Representing Angles: 1. Understanding angles from a static perspective: An angle is a figure formed by two rays originating from a single point. 2. Understanding angles from a dynamic perspective: When a ray rotates around its vertex to another position, the figure formed by these two rays is called an angle. Two rays with a common endpoint form an angle, this common endpoint is called the vertex of the angle, and the two rays are called the sides of the angle. 3. Since rays extend infinitely in one direction, the length of the sides of an angle is irrelevant to the size of the angle. 4. The size of an angle can be measured and compared. 5. Straight angle: A 180° angle. When the two sides of an angle are on the same line, the angle formed is called a straight angle. Specifically, when the ray OA rotates around point O, and the terminal side is on the extension line of the initial side OA in the opposite direction, it forms a straight angle. 6. Right angle: A 90° angle. When the ray OA rotates around point O, and the terminal side is perpendicular to the initial side, it forms a right angle. Half of a straight angle is called a right angle. 7. Acute angle: An angle greater than 0° and less than 90°. An angle smaller than a right angle is called an acute angle. 8. Obtuse angle: An angle greater than 90° and less than 180°. An angle greater than a right angle and less than a straight angle is called an obtuse angle. 9. Full angle: A 360° angle. When the ray OA rotates around point O, and the terminal side coincides with the initial side, it forms a full angle.

3stpes_68

testmini

Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters)

The volume of each small cube is shown in the diagram. What is the volume of the small cube in cubic centimeters?

A. 10; B. 1; C. 0.1; D. No correct answer

B

3steps_1

Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters): 1. Conversion rates between volume units: 1 cubic meter (m³) = 1000 cubic decimeters (dm³); 1 cubic decimeter (dm³) = 1000 cubic centimeters (cm³); 1 cubic decimeter (dm³) = 1 liter (L); 1 cubic centimeter (cm³) = 1 milliliter (mL) 2. Converting between Different Units: To convert from a larger unit to a smaller unit, multiply by the conversion rate. To convert from a smaller unit to a larger unit, divide by the conversion rate. 3. To convert cubic meters to cubic decimeters: multiply by 1000. 4. To convert cubic decimeters to cubic centimeters: multiply by 1000. 5. To convert cubic decimeters to liters: 1 cubic decimeter equals 1 liter. 6. To convert cubic centimeters to milliliters: 1 cubic centimeter equals 1 milliliter. 7. To convert milliliters to cubic centimeters: 1 milliliter equals 1 cubic centimeter. 8. To convert liters to cubic decimeters: 1 liter equals 1 cubic decimeter. 9. To convert cubic decimeters to cubic meters: divide by 1000. 10. To convert cubic centimeters to cubic decimeters: divide by 1000. 11. To convert cubic decimeters to cubic meters: divide by 1000.

3stpes_68

testmini

Observing Objects

As shown in the figure, the volume of each small cube is 1 cubic centimeter. What are the length, width, and height of the rectangular cuboid in centimeters?

A. 6, 5, 4; B. 4, 5, 4; C. 6, 5, 3; D. No correct answer

A

3steps_2

Observing Objects: 1. When observing an object from different angles, you will see two or three adjacent faces, and the shape observed on each face is different. 2. The top, front, and side faces of a rectangular cubiod or cube are defined by convention. The face facing upward is called the top face, the face facing the observer is called the front face, and the two side faces are called the side faces. It is impossible to see opposite faces of a rectangular cubiod or cube at the same time. 3. When observing a combination of objects, not only should the shape of each object be considered, but also the positional relationships between the objects. If an object is blocked by another object, the blocked part will be difficult to observe. 4. No matter which face a cube or sphere is observed from, the shape seen is always the same.

3stpes_68

testmini

Volume and Capacity of Rectangular Cuboids

Given that the length, width, and height of the rectangular cuboid in the figure below, find the volume of the rectangular cuboid ( )in cm³.

A. 90; B. 100; C. 120; D. No correct answer

C

3steps_3

Volume and Capacity of Rectangular Cuboids: 1. The formula for calculating the volume of a rectangular cubiod: The volume of a rectangular cubed = length × width × height. 2. In terms of letters: If V represents the volume of the rectangular cubiod, and a, b, h represent the length, width, and height of the rectangular cubed respectively, then the formula for the volume of the rectangular cubed can be written as: V = abh. 3. The volume of a rectangular cubiod is equal to the base area multiplied by the height. 4. The capacity of a rectangular cubiod usually refers to the amount of space a rectangular container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_68

testmini

Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters);Observing Objects;Volume and Capacity of Rectangular Cuboids

As shown in the figure, the volume of each small cube is 0.001 cubic decimeters. What is the volume of the large rectangular cuboid in cubic centimeters?

A. 90; B. 100; C. 120; D. No correct answer

C

3steps_multi

Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters): 1. Conversion rates between volume units: 1 cubic meter (m³) = 1000 cubic decimeters (dm³); 1 cubic decimeter (dm³) = 1000 cubic centimeters (cm³); 1 cubic decimeter (dm³) = 1 liter (L); 1 cubic centimeter (cm³) = 1 milliliter (mL) 2. Converting between Different Units: To convert from a larger unit to a smaller unit, multiply by the conversion rate. To convert from a smaller unit to a larger unit, divide by the conversion rate. 3. To convert cubic meters to cubic decimeters: multiply by 1000. 4. To convert cubic decimeters to cubic centimeters: multiply by 1000. 5. To convert cubic decimeters to liters: 1 cubic decimeter equals 1 liter. 6. To convert cubic centimeters to milliliters: 1 cubic centimeter equals 1 milliliter. 7. To convert milliliters to cubic centimeters: 1 milliliter equals 1 cubic centimeter. 8. To convert liters to cubic decimeters: 1 liter equals 1 cubic decimeter. 9. To convert cubic decimeters to cubic meters: divide by 1000. 10. To convert cubic centimeters to cubic decimeters: divide by 1000. 11. To convert cubic decimeters to cubic meters: divide by 1000. Observing Objects: 1. When observing an object from different angles, you will see two or three adjacent faces, and the shape observed on each face is different. 2. The top, front, and side faces of a rectangular cubiod or cube are defined by convention. The face facing upward is called the top face, the face facing the observer is called the front face, and the two side faces are called the side faces. It is impossible to see opposite faces of a rectangular cubiod or cube at the same time. 3. When observing a combination of objects, not only should the shape of each object be considered, but also the positional relationships between the objects. If an object is blocked by another object, the blocked part will be difficult to observe. 4. No matter which face a cube or sphere is observed from, the shape seen is always the same. Volume and Capacity of Rectangular Cuboids: 1. The formula for calculating the volume of a rectangular cubiod: The volume of a rectangular cubed = length × width × height. 2. In terms of letters: If V represents the volume of the rectangular cubiod, and a, b, h represent the length, width, and height of the rectangular cubed respectively, then the formula for the volume of the rectangular cubed can be written as: V = abh. 3. The volume of a rectangular cubiod is equal to the base area multiplied by the height. 4. The capacity of a rectangular cubiod usually refers to the amount of space a rectangular container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_69

testmini

Understanding Length (Using a Ruler)

As shown in the figure, using a ruler to measure the length of the rectangular cuboid is () cm.

A. 10; B. 5; C. 4; D. 8; E. No correct answer

A

3steps_1

Understanding Length (Using a Ruler): 1. Length is usually measured using a ruler. 2. When using a ruler for measurement, first determine the measurement unit of the ruler. If there is no special indication, the default unit is centimeters. 3. If one end of the line segment being measured is aligned with the 0 mark on the ruler, the length of the line segment is the direct reading from the other end. 4. If one end of the line segment being measured is not aligned with the 0 mark, the length of the line segment is the difference between the readings at both ends.

3stpes_69

testmini

Combining and Dividing Solids

If the rectangular cuboid in the figure below is cut into the largest possible cube, what is the side length of this cube? (Unit: cm)

A. 10; B. 5; C. 3; D. 4; E. No correct answer

D

3steps_2

Combining and Dividing Solids: 1. When two or more geometric solids are combined to form a new solid, the volume of the new solid equals the sum of the volumes of the original solids. However, the surface area of the new solid is less than the sum of the surface areas of the original solids. If the overlapping area is S, the decrease in surface area is 2S. 2. When a geometric solid is cut into parts, the sum of the volumes of the parts equals the volume of the original solid. However, the sum of the surface areas of the parts is greater than the surface area of the original solid. If the cutting area is S, the increase in surface area is 2S.

3stpes_69

testmini

Volume and Capacity of Cubes

If the rectangular cuboid in the diagram is cut into the largest possible cube, what is the volume of that cube in cubic centimeters? ( )

A. 64; B. 125; C. 1000; D. No correct answer

A

3steps_3

Volume and Capacity of Cubes: 1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length. 2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³. 3. The volume of a cube is equal to the base area multiplied by the height. 4. The capacity of a cube usually refers to the amount of space a cubic container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_69

testmini

Understanding Length (Using a Ruler);Combining and Dividing Solids;Volume and Capacity of Cubes

As shown in the figure, use a ruler to measure the length of the rectangular cuboid, and then cut the rectangular cuboid into a cube with the largest possible volume. What is the volume of this cube in cm³( )? (Unit: cm)

A. 64; B. 125; C. 1000; D. No correct answer

A

3steps_multi

Understanding Length (Using a Ruler): 1. Length is usually measured using a ruler. 2. When using a ruler for measurement, first determine the measurement unit of the ruler. If there is no special indication, the default unit is centimeters. 3. If one end of the line segment being measured is aligned with the 0 mark on the ruler, the length of the line segment is the direct reading from the other end. 4. If one end of the line segment being measured is not aligned with the 0 mark, the length of the line segment is the difference between the readings at both ends. Combining and Dividing Solids: 1. When two or more geometric solids are combined to form a new solid, the volume of the new solid equals the sum of the volumes of the original solids. However, the surface area of the new solid is less than the sum of the surface areas of the original solids. If the overlapping area is S, the decrease in surface area is 2S. 2. When a geometric solid is cut into parts, the sum of the volumes of the parts equals the volume of the original solid. However, the sum of the surface areas of the parts is greater than the surface area of the original solid. If the cutting area is S, the increase in surface area is 2S. Volume and Capacity of Cubes: 1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length. 2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³. 3. The volume of a cube is equal to the base area multiplied by the height. 4. The capacity of a cube usually refers to the amount of space a cubic container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_70

testmini

Surface Area of Cubes

As shown in the figure, a cube with a surface area of 24 cm² is cut off from a rectangular cuboid. What is the edge length of the small cube in cm?

A. 4; B. 6; C. 2; D. No correct answer

C

3steps_1

Surface Area of Cubes: 1. Surface Area of a Cube: The total area of the 6 faces of a cube is its surface area, S = 6s. 2. Method of Calculating the Surface Area of a Cube: Surface area of a cube = edge length × edge length × 6. 3. S = 6*a*a, where S represents the surface area of the cube, and a represents the edge length of the cube.

3stpes_70

testmini

Volume and Capacity of Cubes

As shown in the figure, the volume of the small cube that has been cut off is ( ) (unit: cm³)

A. 20 cm³; B. 8 cm³; C. 40 cm³; D. No correct answer

B

3steps_2

Volume and Capacity of Cubes: 1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length. 2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³. 3. The volume of a cube is equal to the base area multiplied by the height. 4. The capacity of a cube usually refers to the amount of space a cubic container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_70

testmini

Volume and Capacity of Rectangular Cuboids

As shown in the figure, after cutting off a small cube with a volume of 8 cubic centimeters from a rectangular cuboid, the remaining volume is ( )

A. 400 cubic centimeters; B. 8 cubic centimeters; C. 392 cubic centimeters; D. No correct answer

C

3steps_3

Volume and Capacity of Rectangular Cuboids: 1. The formula for calculating the volume of a rectangular cubiod: The volume of a rectangular cubed = length × width × height. 2. In terms of letters: If V represents the volume of the rectangular cubiod, and a, b, h represent the length, width, and height of the rectangular cubed respectively, then the formula for the volume of the rectangular cubed can be written as: V = abh. 3. The volume of a rectangular cubiod is equal to the base area multiplied by the height. 4. The capacity of a rectangular cubiod usually refers to the amount of space a rectangular container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_70

testmini

Surface Area of Cubes;Volume and Capacity of Cubes;Volume and Capacity of Rectangular Cuboids

As shown in the figure, a cube with a surface area of 24 cm² has been cut off. The remaining volume is ( ).

A. 400 cubic centimeters; B. 8 cubic centimeters; C. 392 cubic centimeters; D. No correct answer

C

3steps_multi

Surface Area of Cubes: 1. Surface Area of a Cube: The total area of the 6 faces of a cube is its surface area, S = 6s. 2. Method of Calculating the Surface Area of a Cube: Surface area of a cube = edge length × edge length × 6. 3. S = 6*a*a, where S represents the surface area of the cube, and a represents the edge length of the cube. Volume and Capacity of Cubes: 1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length. 2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³. 3. The volume of a cube is equal to the base area multiplied by the height. 4. The capacity of a cube usually refers to the amount of space a cubic container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods. Volume and Capacity of Rectangular Cuboids: 1. The formula for calculating the volume of a rectangular cubiod: The volume of a rectangular cubed = length × width × height. 2. In terms of letters: If V represents the volume of the rectangular cubiod, and a, b, h represent the length, width, and height of the rectangular cubed respectively, then the formula for the volume of the rectangular cubed can be written as: V = abh. 3. The volume of a rectangular cubiod is equal to the base area multiplied by the height. 4. The capacity of a rectangular cubiod usually refers to the amount of space a rectangular container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_71

testmini

Understanding Sectors

The white part in the figure is composed of 4 identical sectors. These four sectors can be combined to form ( )

A. A semicircle; B. A circle; C. A sector that is neither a semicircle nor a circle; D. No correct answer

B

3steps_1

Understanding Sectors: 1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle. 2. All radii in a sector are equal in length. 3. The part of the circle between two points A and B is called an "arc". 4. An angle with its vertex at the center of the circle is called a "central angle".

3stpes_71

testmini

Area of Circles

The white area in the figure is composed of 4 identical sectors, which can be combined to form a complete circle. What is the area of the white part in cm²? (Unit: cm)

A. π; B. 3π; C. 9π; D. 36π; E. No correct answer

C

3steps_2

Area of Circles: 1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )² 2. Generally, π is taken as 3.14.

3stpes_71

testmini

Area of Squares

The white part in the figure is composed of 4 identical sectors, and the area of the white part is (9π) cm². What is the area of the shaded part ( ) cm²? (Unit: cm)

A. 6 - 3π; B. 36 - 3π; C. 6 - 9π; D. 36 - 9π; E. No correct answer

D

3steps_3

Area of Squares: 1. The area of a square is equal to the square of its side length: S = a * a. 2. The area of a square is equal to the square of the length of its diagonal divided by two.

3stpes_71

testmini

Understanding Sectors;Area of Circles;Area of Squares

The white part in the figure is composed of 4 identical sectors. The area of the shaded part is ( ) cm². (Unit: cm)

A. 6-3π; B. 36-3π; C. 6-9π; D. 36-9π; E. No correct answer

D

3steps_multi

Understanding Sectors: 1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle. 2. All radii in a sector are equal in length. 3. The part of the circle between two points A and B is called an "arc". 4. An angle with its vertex at the center of the circle is called a "central angle". Area of Circles: 1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )² 2. Generally, π is taken as 3.14. Area of Squares: 1. The area of a square is equal to the square of its side length: S = a * a. 2. The area of a square is equal to the square of the length of its diagonal divided by two.

3stpes_72

testmini

Perimeter of Squares

Using a piece of wire, you can form the following square ①. The perimeter of the square is () cm.

A. 5; B. 20; C. 10; D. No correct answer

B

3steps_1

Perimeter of Squares: 1. A square has four equal sides, and the perimeter of a square = side length × 4 (C = 4a).

3stpes_72

testmini

Area of Circles

The diameter AB of circle ② is shown in the figure. What is the area of circle ② in square centimeters? ( )(π = 3.14)

A. 314; B. 628; C. 320.28; D. No correct answer

A

3steps_2

Area of Circles: 1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )² 2. Generally, π is taken as 3.14.

3stpes_72

testmini

Area of Rectangles

In the figure below, the area of rectangle ③ is 314 square centimeters. What is the width of the rectangle ( ) cm?(π = 3.14)

A. 3.14; B. 6.28; C. 15.7; D. No correct answer

C

3steps_3

Area of Rectangles: 1. The area of a rectangle is equal to its length × width, expressed as: S = ab.

3stpes_72

testmini

Perimeter of Squares;Area of Circles;Area of Rectangles

Using a piece of wire to form the square below, and the length of the wire is equal to the length of AB and CD, and the areas of figures ② and ③ are equal, what is the width of the rectangle ( ) cm?(π = 3.14)

A. 3.14; B. 6.28; C. 15.7; D. No correct answer

C

3steps_multi

Perimeter of Squares: 1. A square has four equal sides, and the perimeter of a square = side length × 4 (C = 4a). Area of Circles: 1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )² 2. Generally, π is taken as 3.14. Area of Rectangles: 1. The area of a rectangle is equal to its length × width, expressed as: S = ab.

3stpes_73

testmini

Circumference of Circles

As shown in the figure, the diameter L of the semicircle is close to the circumference of the small circle on the left. Which of the following lengths is closest to L? (π=3.14)

A. 40; B. 30; C. 50; D. No correct answer

A

3steps_1

Circumference of Circles: 1. A line segment passing through the center of a circle with both endpoints on the circle is called the diameter, usually denoted by the letter d. A line segment connecting the center of the circle to any point on the circle is called the radius, usually denoted by the letter r. 2. The circumference of a circle = diameter × pi (C = πd) or the circumference of a circle = 2 × radius × pi (C = 2πr). 3. The circumference of a semicircle can be calculated using the formula C = πr + 2r. 4. In the same circle or congruent circles, there are an infinite number of radii and an infinite number of diameters. All radii and diameters in the same circle are equal. The length of the diameter is twice the length of the radius, and the length of the radius is half the length of the diameter.

3stpes_73

testmini

Understanding Circles

As shown in the figure, a 4-meter-wide path is laid along the outer edge of a semicircular lawn. What is the radius of the circle composed of the path and the lawn? () meters.

A. 16; B. 24; C. 20; D. No correct answer

B

3steps_2

Understanding Circles: 1. In a plane, a circle is defined as the set of all points that are at a fixed distance from a fixed point. 2. A circle is an axisymmetric figure. 3. The center of the circle is called the center, usually denoted by the letter O. A line segment connecting the center to any point on the circle is called the radius, usually denoted by the letter r. 4. A line segment that passes through the center and has its endpoints on the circle is called the diameter, usually denoted by the letter d. The diameter is the longest line segment within the circle, and it is twice the length of the radius. 5. All radii of a circle are equal in length, and all diameters are equal in length. 6. Method of drawing a circle: Use a compass. The distance between the legs of the compass represents the radius. After setting the center and the radius, rotate the compass to draw a circle.

3stpes_73

testmini

Area of Circles

As shown in the figure, a 4-meter-wide path is laid along the perimeter of a semicircular lawn. Given that the radius of the path is 24 meters, which of the following formulas should be used to find the area of the path? (π=3.14)

A. 3.14×(24^2－20^2)÷2; B. 3.14×(24^2－20^2); C. 3.14×(22^2－20^2)÷2; D. 3.14×(22^2－20^2); E. No correct answer

A

3steps_3

Area of Circles: 1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )² 2. Generally, π is taken as 3.14.

3stpes_73

testmini

Circumference of Circles;Understanding Circles;Area of Circles

As shown in the figure, the diameter L of the semicircular lawn is close to the circumference of the small circle on the left. A 4-meter-wide path is laid around the semicircular lawn. Which of the following formulas should be used to find the area of the path?(Use π = 3.14)

A. 3.14×(24^2－20^2)÷2; B. 3.14×(24^2－20^2); C. 3.14×(22^2－20^2)÷2; D. 3.14×(22^2－20^2); E. No correct answer

A

3steps_multi

Circumference of Circles: 1. A line segment passing through the center of a circle with both endpoints on the circle is called the diameter, usually denoted by the letter d. A line segment connecting the center of the circle to any point on the circle is called the radius, usually denoted by the letter r. 2. The circumference of a circle = diameter × pi (C = πd) or the circumference of a circle = 2 × radius × pi (C = 2πr). 3. The circumference of a semicircle can be calculated using the formula C = πr + 2r. 4. In the same circle or congruent circles, there are an infinite number of radii and an infinite number of diameters. All radii and diameters in the same circle are equal. The length of the diameter is twice the length of the radius, and the length of the radius is half the length of the diameter. Understanding Circles: 1. In a plane, a circle is defined as the set of all points that are at a fixed distance from a fixed point. 2. A circle is an axisymmetric figure. 3. The center of the circle is called the center, usually denoted by the letter O. A line segment connecting the center to any point on the circle is called the radius, usually denoted by the letter r. 4. A line segment that passes through the center and has its endpoints on the circle is called the diameter, usually denoted by the letter d. The diameter is the longest line segment within the circle, and it is twice the length of the radius. 5. All radii of a circle are equal in length, and all diameters are equal in length. 6. Method of drawing a circle: Use a compass. The distance between the legs of the compass represents the radius. After setting the center and the radius, rotate the compass to draw a circle. Area of Circles: 1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )² 2. Generally, π is taken as 3.14.

3stpes_74

testmini

Volume and Capacity of Cubes

As shown in the figure, the volume of the small cube is 1 cm³. What is the edge length of the small cube? ( )

A. 4; B. 3; C. 2; D. 1; E. No correct answer

D

3steps_1

Volume and Capacity of Cubes: 1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length. 2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³. 3. The volume of a cube is equal to the base area multiplied by the height. 4. The capacity of a cube usually refers to the amount of space a cubic container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.

3stpes_74

testmini

Observing Objects

As shown in the figure, multiple small cubes with a side length of 1 cm are combined to form a large rectangular cuboid. The length, width, and height of this rectangular cuboid are respectively () cm.

A. 4, 3, 2; B. 3, 2, 1; C. 3, 3, 2; D. 4, 2, 1; E. No correct answer

B

3steps_2

Observing Objects: 1. When observing an object from different angles, you will see two or three adjacent faces, and the shape observed on each face is different. 2. The top, front, and side faces of a rectangular cubiod or cube are defined by convention. The face facing upward is called the top face, the face facing the observer is called the front face, and the two side faces are called the side faces. It is impossible to see opposite faces of a rectangular cubiod or cube at the same time. 3. When observing a combination of objects, not only should the shape of each object be considered, but also the positional relationships between the objects. If an object is blocked by another object, the blocked part will be difficult to observe. 4. No matter which face a cube or sphere is observed from, the shape seen is always the same.

3stpes_74

testmini

Surface Area of Rectangular Cuboids

As shown in the figure, multiple small cubes with a side length of 1 cm each are combined to form a large rectangular cuboid. What is the surface area of this rectangular cuboid in cm²?

A. 22; B. 21; C. 20; D. 18; E. No correct answer

A

3steps_3

Surface Area of Rectangular Cuboids: 1. Definition of Surface Area: The total area of the 6 faces of a rectangular cubiod or a cube is called its surface area. 2. Surface area of a rectangular cubiod = length × width × 2 + length × height × 2 + width × height × 2. 3. Surface area of a rectangular cubiod = (length × width + length × height + width × height) × 2. 4. Surface Area of a Rectangular Cubiod: If the letters a, b, and h represent the length, width, and height of a rectangular cubiod respectively, and S represents the surface area of the rectangular cubiod, then S = 2ab + 2ah + 2bh or S = 2(ab + ah + bh).

3stpes_74

testmini

Volume and Capacity of Cubes;Observing Objects;Surface Area of Rectangular Cuboids

As shown in the figure, multiple small cubes with a volume of 1 cm³ each are combined to form a large rectangular cuboid. What is the surface area of this rectangular cuboid?

A. 22; B. 21; C. 20; D. 18; E. No correct answer

A

3steps_multi

Volume and Capacity of Cubes: 1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length. 2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³. 3. The volume of a cube is equal to the base area multiplied by the height. 4. The capacity of a cube usually refers to the amount of space a cubic container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods. Observing Objects: 1. When observing an object from different angles, you will see two or three adjacent faces, and the shape observed on each face is different. 2. The top, front, and side faces of a rectangular cubiod or cube are defined by convention. The face facing upward is called the top face, the face facing the observer is called the front face, and the two side faces are called the side faces. It is impossible to see opposite faces of a rectangular cubiod or cube at the same time. 3. When observing a combination of objects, not only should the shape of each object be considered, but also the positional relationships between the objects. If an object is blocked by another object, the blocked part will be difficult to observe. 4. No matter which face a cube or sphere is observed from, the shape seen is always the same. Surface Area of Rectangular Cuboids: 1. Definition of Surface Area: The total area of the 6 faces of a rectangular cubiod or a cube is called its surface area. 2. Surface area of a rectangular cubiod = length × width × 2 + length × height × 2 + width × height × 2. 3. Surface area of a rectangular cubiod = (length × width + length × height + width × height) × 2. 4. Surface Area of a Rectangular Cubiod: If the letters a, b, and h represent the length, width, and height of a rectangular cubiod respectively, and S represents the surface area of the rectangular cubiod, then S = 2ab + 2ah + 2bh or S = 2(ab + ah + bh).

3stpes_75

testmini

Properties and Understanding of Trapezoids

As shown in the figure, quadrilateral OABC is a right trapezoid. Then ∠OAB = ∠ABC = ( )°

A. 45; B. 60; C. 72; D. 90; E. No correct answer

D

3steps_1

Properties and Understanding of Trapezoids: 1. A trapezoid (or trapezium) is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called the bases of the trapezoid: the longer base is called the lower base, and the shorter base is called the upper base; the other two sides are called the legs; the perpendicular segment between the two bases is called the height of the trapezoid. 2. A trapezoid with one leg perpendicular to the bases is called a right trapezoid. 3. A trapezoid with both legs equal in length is called an isosceles trapezoid. 4. The height of a trapezoid is the distance between the upper base and the lower base.

3stpes_75

testmini

Tessellation of Figures

In the tiling pattern shown in the figure, which is composed of identical right-angled trapezoids, the angle ∠AOC at the junction is equal to ( )°.

A. 45; B. 60; C. 72; D. 90; E. No correct answer

A

3steps_2

Tessellation of Figures: 1. Tiling, or tessellation of a plane, refers to the process of covering a plane using one or more types of shapes that are identical in size and shape, without leaving any gaps or overlapping. 2. Common shapes for tiling: square, rectangle, triangle, parallelogram, regular hexagon. 3. The sum of the angles at each vertex point of the tiling must equal a full angle (360°). A plane can only be tiled if the sum of the angles at each vertex point equals a full angle. 4. Since the sum of the interior angles of a triangle is 180°, using several congruent triangles, each angle can be used twice to form a full angle, so triangles can always tile a plane. 5. The sum of the interior angles of any quadrilateral is 360°, and at each vertex point of the tiling, the four angles just form a full angle, so any quadrilateral can tile a plane. 6. Each interior angle of a regular hexagon is 120°, and they can also form a full angle, so regular hexagons can tile a plane.

3stpes_75

testmini

Sum of Interior Angles of Other Polygons

In the quadrilateral OABC shown in the figure, what is the measure of ∠OCB?

A. 135; B. 130; C. 120; D. 90; E. No correct answer

A

3steps_3

Sum of Interior Angles of Other Polygons: 1. The sum of the interior angles of a quadrilateral is 360°, and the sum of the interior angles of a pentagon is 540°. 2. The sum of the interior angles of a polygon = 180° × (number of sides - 2). 3. The sum of the interior angles of an n-sided polygon = (n - 2) × 180°. 4. A polygon with all sides of equal length and all interior angles equal is called a regular polygon.

3stpes_75

testmini

Properties and Understanding of Trapezoids;Tessellation of Figures;Sum of Interior Angles of Other Polygons

As shown in the figure, quadrilateral OABC is a right trapezoid. The 8 right trapezoids in the figure are completely identical and form a tessellation pattern. What is the measure of ∠OCB? ( ) degrees

A. 135; B. 130; C. 120; D. 90; E. No correct answer

A

3steps_multi

Properties and Understanding of Trapezoids: 1. A trapezoid (or trapezium) is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called the bases of the trapezoid: the longer base is called the lower base, and the shorter base is called the upper base; the other two sides are called the legs; the perpendicular segment between the two bases is called the height of the trapezoid. 2. A trapezoid with one leg perpendicular to the bases is called a right trapezoid. 3. A trapezoid with both legs equal in length is called an isosceles trapezoid. 4. The height of a trapezoid is the distance between the upper base and the lower base. Tessellation of Figures: 1. Tiling, or tessellation of a plane, refers to the process of covering a plane using one or more types of shapes that are identical in size and shape, without leaving any gaps or overlapping. 2. Common shapes for tiling: square, rectangle, triangle, parallelogram, regular hexagon. 3. The sum of the angles at each vertex point of the tiling must equal a full angle (360°). A plane can only be tiled if the sum of the angles at each vertex point equals a full angle. 4. Since the sum of the interior angles of a triangle is 180°, using several congruent triangles, each angle can be used twice to form a full angle, so triangles can always tile a plane. 5. The sum of the interior angles of any quadrilateral is 360°, and at each vertex point of the tiling, the four angles just form a full angle, so any quadrilateral can tile a plane. 6. Each interior angle of a regular hexagon is 120°, and they can also form a full angle, so regular hexagons can tile a plane. Sum of Interior Angles of Other Polygons: 1. The sum of the interior angles of a quadrilateral is 360°, and the sum of the interior angles of a pentagon is 540°. 2. The sum of the interior angles of a polygon = 180° × (number of sides - 2). 3. The sum of the interior angles of an n-sided polygon = (n - 2) × 180°. 4. A polygon with all sides of equal length and all interior angles equal is called a regular polygon.