Chapter 49: Congruence and Similarity
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Congruence and Similarity with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Factor (a number that divides another number exactly)
- Surface Area (total area of the outside of a 3D object)
- Estimate (a close approximation used to check whether an answer is reasonable)
- Solution (a value or result that satisfies the problem)
- Representation (a way to show mathematics using symbols, diagrams, tables, graphs, or words)
- Reasonableness (whether an answer makes sense in the context of the problem)
How to Learn This Chapter
Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
49.1 Congruent figures
Congruent figures have the same size and shape. This topic focuses on Congruent figures .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Congruent figures: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Congruent figures have the same size and shape. This topic focuses on Congruent figures .
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Congruent figures: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Congruent figures have the same size and shape. This topic focuses on Congruent figures .
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Congruent figures: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Congruent figures have the same size and shape. This topic focuses on Congruent figures .
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Congruent figures: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Congruent figures have the same size and shape. This topic focuses on Congruent figures .
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Congruent figures: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Congruent figures have the same size and shape. This topic focuses on Congruent figures .
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Congruent figures. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.2 Corresponding sides
Corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 10 cm
Worked Example 6
Problem: Explain Corresponding sides in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Corresponding sides and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Corresponding sides using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Corresponding sides problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Corresponding sides could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Corresponding sides. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.3 Corresponding angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Corresponding angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Corresponding angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Corresponding angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Corresponding angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Corresponding angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Corresponding angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Corresponding angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.4 Similar figures
Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Similar figures .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Similar figures .
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Similar figures .
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Similar figures .
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Similar figures .
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Similar figures .
Answer: 10 cm
Worked Example 6
Problem: An original length is 8 cm and the scale factor is 0.5. Find the image length.
- Multiply the original length by the scale factor.
- 8 × 0.5 = 4.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 4 cm
Worked Example 7
Problem: An original length is 12 cm and the scale factor is 1.5. Find the image length.
- Multiply the original length by the scale factor.
- 12 × 1.5 = 18.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 18 cm
Worked Example 8
Problem: An original length is 19 cm and the scale factor is 2. Find the image length.
- Multiply the original length by the scale factor.
- 19 × 2 = 38.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 38 cm
Worked Example 9
Problem: An original length is 22 cm and the scale factor is 2.5. Find the image length.
- Multiply the original length by the scale factor.
- 22 × 2.5 = 55.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 55 cm
Worked Example 10
Problem: An original length is 25 cm and the scale factor is 3. Find the image length.
- Multiply the original length by the scale factor.
- 25 × 3 = 75.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 75 cm
Practice Exercise
Create one new question about Similar figures. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.5 Scale factor
A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: List the factors of 18.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor .
Answer: 1, 2, 3, 6, 9, 18
Worked Example 2
Problem: List the factors of 24.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor .
Answer: 1, 2, 3, 4, 6, 8, 12, 24
Worked Example 3
Problem: List the factors of 36.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor .
Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36
Worked Example 4
Problem: List the factors of 45.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor .
Answer: 1, 3, 5, 9, 15, 45
Worked Example 5
Problem: List the factors of 60.
- Find all whole numbers that divide exactly.
Very beginner explanation: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor .
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Worked Example 6
Problem: List all positive factors of 18.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 6, 9, 18
Worked Example 7
Problem: List all positive factors of 24.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 4, 6, 8, 12, 24
Worked Example 8
Problem: List all positive factors of 30.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 5, 6, 10, 15, 30
Worked Example 9
Problem: List all positive factors of 42.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 6, 7, 14, 21, 42
Worked Example 10
Problem: List all positive factors of 60.
- Test whole numbers that divide with no remainder.
Very beginner explanation: A factor divides a number exactly.
Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Practice Exercise
Create one new question about Scale factor. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.6 Similar triangles introduction
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Similar triangles introduction .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Similar triangles introduction .
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Similar triangles introduction .
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Similar triangles introduction .
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Similar triangles introduction .
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Similar triangles introduction .
Answer: 10 cm
Worked Example 6
Problem: An original length is 8 cm and the scale factor is 0.5. Find the image length.
- Multiply the original length by the scale factor.
- 8 × 0.5 = 4.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 4 cm
Worked Example 7
Problem: An original length is 12 cm and the scale factor is 1.5. Find the image length.
- Multiply the original length by the scale factor.
- 12 × 1.5 = 18.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 18 cm
Worked Example 8
Problem: An original length is 19 cm and the scale factor is 2. Find the image length.
- Multiply the original length by the scale factor.
- 19 × 2 = 38.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 38 cm
Worked Example 9
Problem: An original length is 22 cm and the scale factor is 2.5. Find the image length.
- Multiply the original length by the scale factor.
- 22 × 2.5 = 55.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 55 cm
Worked Example 10
Problem: An original length is 25 cm and the scale factor is 3. Find the image length.
- Multiply the original length by the scale factor.
- 25 × 3 = 75.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 75 cm
Practice Exercise
Create one new question about Similar triangles introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.7 Finding missing corresponding sides
Finding missing corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Finding missing corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Finding missing corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Finding missing corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Finding missing corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Finding missing corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 10 cm
Worked Example 6
Problem: Explain Finding missing corresponding sides in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding missing corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Finding missing corresponding sides and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding missing corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Finding missing corresponding sides using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding missing corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Finding missing corresponding sides problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding missing corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Finding missing corresponding sides could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Finding missing corresponding sides becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Finding missing corresponding sides. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.8 Finding missing angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Finding missing angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Finding missing angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Finding missing angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Finding missing angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Finding missing angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Finding missing angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Finding missing angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.9 Perimeter of similar figures
Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Perimeter of similar figures .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Perimeter of similar figures .
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Perimeter of similar figures .
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Perimeter of similar figures .
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Perimeter of similar figures .
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Perimeter of similar figures .
Answer: 10 cm
Worked Example 6
Problem: An original length is 8 cm and the scale factor is 0.5. Find the image length.
- Multiply the original length by the scale factor.
- 8 × 0.5 = 4.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 4 cm
Worked Example 7
Problem: An original length is 12 cm and the scale factor is 1.5. Find the image length.
- Multiply the original length by the scale factor.
- 12 × 1.5 = 18.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 18 cm
Worked Example 8
Problem: An original length is 19 cm and the scale factor is 2. Find the image length.
- Multiply the original length by the scale factor.
- 19 × 2 = 38.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 38 cm
Worked Example 9
Problem: An original length is 22 cm and the scale factor is 2.5. Find the image length.
- Multiply the original length by the scale factor.
- 22 × 2.5 = 55.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 55 cm
Worked Example 10
Problem: An original length is 25 cm and the scale factor is 3. Find the image length.
- Multiply the original length by the scale factor.
- 25 × 3 = 75.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 75 cm
Practice Exercise
Create one new question about Perimeter of similar figures. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.10 Area comparison introduction
Area comparison introduction is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Area comparison introduction: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Area comparison introduction is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Area comparison introduction: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Area comparison introduction is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Area comparison introduction: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Area comparison introduction is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Area comparison introduction: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Area comparison introduction is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Area comparison introduction: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Area comparison introduction is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 6
Problem: Find the area of a rectangle 6 cm by 3 cm.
- Use A = length × width.
- A = 6 × 3.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 18 cm²
Worked Example 7
Problem: Find the area of a rectangle 7 cm by 4 cm.
- Use A = length × width.
- A = 7 × 4.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 28 cm²
Worked Example 8
Problem: Find the area of a rectangle 8 cm by 5 cm.
- Use A = length × width.
- A = 8 × 5.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 40 cm²
Worked Example 9
Problem: Find the area of a rectangle 9 cm by 6 cm.
- Use A = length × width.
- A = 9 × 6.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 54 cm²
Worked Example 10
Problem: Find the area of a rectangle 10 cm by 7 cm.
- Use A = length × width.
- A = 10 × 7.
Very beginner explanation: Area measures the 2D space inside a figure and uses square units.
Answer: 70 cm²
Practice Exercise
Create one new question about Area comparison introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.11 Checking whether figures are similar
Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Checking whether figures are similar .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Checking whether figures are similar .
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Checking whether figures are similar .
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Checking whether figures are similar .
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Checking whether figures are similar .
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Checking whether figures are similar .
Answer: 10 cm
Worked Example 6
Problem: An original length is 8 cm and the scale factor is 0.5. Find the image length.
- Multiply the original length by the scale factor.
- 8 × 0.5 = 4.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 4 cm
Worked Example 7
Problem: An original length is 12 cm and the scale factor is 1.5. Find the image length.
- Multiply the original length by the scale factor.
- 12 × 1.5 = 18.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 18 cm
Worked Example 8
Problem: An original length is 19 cm and the scale factor is 2. Find the image length.
- Multiply the original length by the scale factor.
- 19 × 2 = 38.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 38 cm
Worked Example 9
Problem: An original length is 22 cm and the scale factor is 2.5. Find the image length.
- Multiply the original length by the scale factor.
- 22 × 2.5 = 55.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 55 cm
Worked Example 10
Problem: An original length is 25 cm and the scale factor is 3. Find the image length.
- Multiply the original length by the scale factor.
- 25 × 3 = 75.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 75 cm
Practice Exercise
Create one new question about Checking whether figures are similar. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
49.12 Real-life similarity
Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Real-life similarity .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Real-life similarity .
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Real-life similarity .
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Real-life similarity .
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Real-life similarity .
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Real-life similarity .
Answer: 10 cm
Worked Example 6
Problem: An original length is 8 cm and the scale factor is 0.5. Find the image length.
- Multiply the original length by the scale factor.
- 8 × 0.5 = 4.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 4 cm
Worked Example 7
Problem: An original length is 12 cm and the scale factor is 1.5. Find the image length.
- Multiply the original length by the scale factor.
- 12 × 1.5 = 18.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 18 cm
Worked Example 8
Problem: An original length is 19 cm and the scale factor is 2. Find the image length.
- Multiply the original length by the scale factor.
- 19 × 2 = 38.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 38 cm
Worked Example 9
Problem: An original length is 22 cm and the scale factor is 2.5. Find the image length.
- Multiply the original length by the scale factor.
- 22 × 2.5 = 55.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 55 cm
Worked Example 10
Problem: An original length is 25 cm and the scale factor is 3. Find the image length.
- Multiply the original length by the scale factor.
- 25 × 3 = 75.
Very beginner explanation: A scale factor multiplies every corresponding length by the same amount.
Answer: 75 cm
Practice Exercise
Create one new question about Real-life similarity. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
Chapter Notes, Practice, and Common Mistakes
Chapter Notes
- Read the complete question before choosing an operation, formula, graph, or model.
- Write technical words together with their meaning until you are comfortable using them.
- Show all important steps so another student can follow your reasoning.
- Keep units, labels, signs, axes, variables, and mathematical symbols clear.
- Estimate before or after calculating when an estimate can help check reasonableness.
- For real-life problems, explain what the final number means in the situation.
Extra Practice
- Create and solve a new question about Congruent figures . Show your reasoning and check your answer.
- Create and solve a new question about Corresponding sides . Show your reasoning and check your answer.
- Create and solve a new question about Corresponding angles . Show your reasoning and check your answer.
- Create and solve a new question about Similar figures . Show your reasoning and check your answer.
- Create and solve a new question about Scale factor . Show your reasoning and check your answer.
- Create and solve a new question about Similar triangles introduction . Show your reasoning and check your answer.
- Create and solve a new question about Finding missing corresponding sides . Show your reasoning and check your answer.
- Create and solve a new question about Finding missing angles . Show your reasoning and check your answer.
- Create and solve a new question about Perimeter of similar figures . Show your reasoning and check your answer.
- Create and solve a new question about Area comparison introduction . Show your reasoning and check your answer.
- Create and solve a new question about Checking whether figures are similar . Show your reasoning and check your answer.
- Create and solve a new question about Real-life similarity . Show your reasoning and check your answer.
Common Mistakes
- Assuming a diagram is drawn to scale.
- Confusing radius and diameter.
- Using square units for volume or cubic units for area.
- Applying a scale factor to only one dimension.
- Forgetting the circular bases in cylinder surface area.
30 Review Questions and Answers
Q1. What is important to remember about Congruent figures?
Answer: Congruent figures have the same size and shape. This topic focuses on Congruent figures.
Q2. What is important to remember about Corresponding sides?
Answer: Corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q3. What is important to remember about Corresponding angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Corresponding angles.
Q4. What is important to remember about Similar figures?
Answer: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Similar figures.
Q5. What is important to remember about Scale factor?
Answer: A factor (a number that divides another number exactly) helps with simplifying, grouping, and fraction work. This topic focuses on Scale factor.
Q6. What is important to remember about Similar triangles introduction?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Similar triangles introduction.
Q7. What is important to remember about Finding missing corresponding sides?
Answer: Finding missing corresponding sides is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q8. What is important to remember about Finding missing angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Finding missing angles.
Q9. What is important to remember about Perimeter of similar figures?
Answer: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Perimeter of similar figures.
Q10. What is important to remember about Area comparison introduction?
Answer: Area comparison introduction is an important Grade 7 idea in Congruence and Similarity. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q11. What is important to remember about Checking whether figures are similar?
Answer: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Checking whether figures are similar.
Q12. What is important to remember about Real-life similarity?
Answer: Similar figures have the same shape and proportional corresponding lengths. This topic focuses on Real-life similarity.
Q13. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q14. Why is estimation useful?
Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.
Q15. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q16. When should you round?
Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.
Q17. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q18. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the relationship connecting them.
Q19. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q20. What should you do if an answer seems unreasonable?
Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.
Q21. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q22. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q23. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q24. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.
Q25. How should you study this chapter?
Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.
Q26. When is a calculator useful?
Answer: A calculator is useful for lengthy arithmetic after you understand the mathematical setup and can estimate the expected size of the answer.
Q27. Why compare more than one strategy?
Answer: Different strategies can make a problem easier and provide a way to verify the result.
Q28. What is a mathematical model?
Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.
Q29. Why should assumptions be stated?
Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.
Q30. Why should sources be checked in data problems?
Answer: Reliable sources and fair collection methods make conclusions more trustworthy.