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Chapter 40: Congruence, Similarity, and Scale

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
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What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

40.1 Congruent figures

Congruent figures have the same size and shape. In this section, the focus is Congruent figures.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Congruent figures: Explain Congruent figures in one simple sentence.

  1. Look at the words in “Congruent figures”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Congruent figures is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Congruent figures: A student says, “I can use Congruent figures without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Congruent figures: What is the first step when solving a problem about Congruent figures?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Congruent figures: After solving a Congruent figures problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Congruent figures: Give one way Congruent figures could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Congruent figures can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Congruent figures: Which representation could help explain Congruent figures: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Congruent figures: A student gets an answer for Congruent figures but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Congruent figures: Why can estimation help before a detailed Congruent figures calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Congruent figures: How can you test whether your rule for Congruent figures works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Congruent figures: How would you teach Congruent figures to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Congruent figures. Show the important steps, include units when needed, and explain how you checked the answer.

40.2 Corresponding sides

Corresponding sides is an important Grade 8 concept in Congruence, Similarity, and Scale. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Corresponding sides: Explain Corresponding sides in one simple sentence.

  1. Look at the words in “Corresponding sides”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Corresponding sides is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Corresponding sides: A student says, “I can use Corresponding sides without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Corresponding sides: What is the first step when solving a problem about Corresponding sides?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Corresponding sides: After solving a Corresponding sides problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Corresponding sides: Give one way Corresponding sides could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Corresponding sides can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Corresponding sides: Which representation could help explain Corresponding sides: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Corresponding sides: A student gets an answer for Corresponding sides but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Corresponding sides: Why can estimation help before a detailed Corresponding sides calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Corresponding sides: How can you test whether your rule for Corresponding sides works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Corresponding sides: How would you teach Corresponding sides to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Corresponding sides. Show the important steps, include units when needed, and explain how you checked the answer.

40.3 Corresponding angles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Corresponding angles.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 35°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 35°

Worked Example 2

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 48°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 48°

Worked Example 3

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 67°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 67°

Worked Example 4

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 72°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 72°

Worked Example 5

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 110°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 110°

Worked Example 6

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 25°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 25°

Worked Example 7

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 58°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 58°

Worked Example 8

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 83°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 83°

Worked Example 9

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 95°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 95°

Worked Example 10

Problem: Two parallel lines are cut by a transversal. One corresponding angle is 120°. Find the matching corresponding angle.

  1. Corresponding angles are equal when the lines are parallel.

Very beginner explanation: Parallel-line angle rules let one known angle determine several others.

Answer: 120°

Practice exercise

Create one new Grade 8 problem involving Corresponding angles. Show the important steps, include units when needed, and explain how you checked the answer.

40.4 Similar figures

Similar figures have the same shape and proportional corresponding lengths. In this section, the focus is Similar figures.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: An original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

Create one new Grade 8 problem involving Similar figures. Show the important steps, include units when needed, and explain how you checked the answer.

40.5 Scale factors

A factor (a whole number that divides another number exactly) helps with simplifying fractions and finding common group sizes. In this section, the focus is Scale factors.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: An original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

Create one new Grade 8 problem involving Scale factors. Show the important steps, include units when needed, and explain how you checked the answer.

40.6 Similar triangles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Similar triangles.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A triangle has angles 35° and 45°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-35-45=100.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 100°

Worked Example 2

Problem: A triangle has angles 48° and 58°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-48-58=74.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 74°

Worked Example 3

Problem: A triangle has angles 67° and 47°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-67-47=66.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 66°

Worked Example 4

Problem: A triangle has angles 72° and 52°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-72-52=56.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 56°

Worked Example 5

Problem: A triangle has angles 110° and 60°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-110-60=10.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 10°

Worked Example 6

Problem: A triangle has angles 25° and 65°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-25-65=90.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 90°

Worked Example 7

Problem: A triangle has angles 58° and 68°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-58-68=54.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 54°

Worked Example 8

Problem: A triangle has angles 83° and 63°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-83-63=34.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 34°

Worked Example 9

Problem: A triangle has angles 95° and 45°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-95-45=40.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 40°

Worked Example 10

Problem: A triangle has angles 120° and 40°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-120-40=20.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 20°

Practice exercise

Create one new Grade 8 problem involving Similar triangles. Show the important steps, include units when needed, and explain how you checked the answer.

40.7 Finding missing sides

Finding missing sides is an important Grade 8 concept in Congruence, Similarity, and Scale. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Finding missing sides: Explain Finding missing sides in one simple sentence.

  1. Look at the words in “Finding missing sides”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Finding missing sides is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Finding missing sides: A student says, “I can use Finding missing sides without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Finding missing sides: What is the first step when solving a problem about Finding missing sides?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Finding missing sides: After solving a Finding missing sides problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Finding missing sides: Give one way Finding missing sides could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Finding missing sides can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Finding missing sides: Which representation could help explain Finding missing sides: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Finding missing sides: A student gets an answer for Finding missing sides but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Finding missing sides: Why can estimation help before a detailed Finding missing sides calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Finding missing sides: How can you test whether your rule for Finding missing sides works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Finding missing sides: How would you teach Finding missing sides to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Finding missing sides. Show the important steps, include units when needed, and explain how you checked the answer.

40.8 Finding missing angles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Finding missing angles.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the supplementary angle to 35°.

  1. Supplementary angles total 180°.
  2. 180° - 35° = 145°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 145°

Worked Example 2

Problem: Find the supplementary angle to 48°.

  1. Supplementary angles total 180°.
  2. 180° - 48° = 132°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 132°

Worked Example 3

Problem: Find the supplementary angle to 67°.

  1. Supplementary angles total 180°.
  2. 180° - 67° = 113°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 113°

Worked Example 4

Problem: Find the supplementary angle to 72°.

  1. Supplementary angles total 180°.
  2. 180° - 72° = 108°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 108°

Worked Example 5

Problem: Find the supplementary angle to 110°.

  1. Supplementary angles total 180°.
  2. 180° - 110° = 70°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 70°

Worked Example 6

Problem: Find the supplementary angle to 25°.

  1. Supplementary angles total 180°.
  2. 180° - 25° = 155°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 155°

Worked Example 7

Problem: Find the supplementary angle to 58°.

  1. Supplementary angles total 180°.
  2. 180° - 58° = 122°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 122°

Worked Example 8

Problem: Find the supplementary angle to 83°.

  1. Supplementary angles total 180°.
  2. 180° - 83° = 97°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 97°

Worked Example 9

Problem: Find the supplementary angle to 95°.

  1. Supplementary angles total 180°.
  2. 180° - 95° = 85°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 85°

Worked Example 10

Problem: Find the supplementary angle to 120°.

  1. Supplementary angles total 180°.
  2. 180° - 120° = 60°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 60°

Practice exercise

Create one new Grade 8 problem involving Finding missing angles. Show the important steps, include units when needed, and explain how you checked the answer.

40.9 Comparing perimeter

Perimeter is the total distance around a two-dimensional figure. In this section, the focus is Comparing perimeter.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the perimeter of a rectangle 5 cm by 2 cm.

  1. Use P = 2(l+w).
  2. P = 2(5+2).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 14 cm

Worked Example 2

Problem: Find the perimeter of a rectangle 6 cm by 3 cm.

  1. Use P = 2(l+w).
  2. P = 2(6+3).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 18 cm

Worked Example 3

Problem: Find the perimeter of a rectangle 7 cm by 4 cm.

  1. Use P = 2(l+w).
  2. P = 2(7+4).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 22 cm

Worked Example 4

Problem: Find the perimeter of a rectangle 8 cm by 5 cm.

  1. Use P = 2(l+w).
  2. P = 2(8+5).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 26 cm

Worked Example 5

Problem: Find the perimeter of a rectangle 9 cm by 6 cm.

  1. Use P = 2(l+w).
  2. P = 2(9+6).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 30 cm

Worked Example 6

Problem: Find the perimeter of a rectangle 10 cm by 7 cm.

  1. Use P = 2(l+w).
  2. P = 2(10+7).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 34 cm

Worked Example 7

Problem: Find the perimeter of a rectangle 11 cm by 8 cm.

  1. Use P = 2(l+w).
  2. P = 2(11+8).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 38 cm

Worked Example 8

Problem: Find the perimeter of a rectangle 12 cm by 9 cm.

  1. Use P = 2(l+w).
  2. P = 2(12+9).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 42 cm

Worked Example 9

Problem: Find the perimeter of a rectangle 13 cm by 10 cm.

  1. Use P = 2(l+w).
  2. P = 2(13+10).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 46 cm

Worked Example 10

Problem: Find the perimeter of a rectangle 14 cm by 11 cm.

  1. Use P = 2(l+w).
  2. P = 2(14+11).

Very beginner explanation: Perimeter is the total distance around the outside of a figure.

Answer: 50 cm

Practice exercise

Create one new Grade 8 problem involving Comparing perimeter. Show the important steps, include units when needed, and explain how you checked the answer.

40.10 Comparing area

Area measures the amount of two-dimensional space inside a figure and uses square units. In this section, the focus is Comparing area.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the area of a rectangle 6 cm by 3 cm.

  1. Use A = length × width.
  2. A = 6×3.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 18 cm²

Worked Example 2

Problem: Find the area of a rectangle 7 cm by 4 cm.

  1. Use A = length × width.
  2. A = 7×4.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 28 cm²

Worked Example 3

Problem: Find the area of a rectangle 8 cm by 5 cm.

  1. Use A = length × width.
  2. A = 8×5.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 40 cm²

Worked Example 4

Problem: Find the area of a rectangle 9 cm by 6 cm.

  1. Use A = length × width.
  2. A = 9×6.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 54 cm²

Worked Example 5

Problem: Find the area of a rectangle 10 cm by 7 cm.

  1. Use A = length × width.
  2. A = 10×7.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 70 cm²

Worked Example 6

Problem: Find the area of a rectangle 11 cm by 8 cm.

  1. Use A = length × width.
  2. A = 11×8.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 88 cm²

Worked Example 7

Problem: Find the area of a rectangle 12 cm by 9 cm.

  1. Use A = length × width.
  2. A = 12×9.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 108 cm²

Worked Example 8

Problem: Find the area of a rectangle 13 cm by 10 cm.

  1. Use A = length × width.
  2. A = 13×10.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 130 cm²

Worked Example 9

Problem: Find the area of a rectangle 14 cm by 11 cm.

  1. Use A = length × width.
  2. A = 14×11.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 154 cm²

Worked Example 10

Problem: Find the area of a rectangle 15 cm by 12 cm.

  1. Use A = length × width.
  2. A = 15×12.

Very beginner explanation: Area measures the space inside a 2D figure and uses square units.

Answer: 180 cm²

Practice exercise

Create one new Grade 8 problem involving Comparing area. Show the important steps, include units when needed, and explain how you checked the answer.

40.11 Scale drawings review

Scale drawings review is an important Grade 8 concept in Congruence, Similarity, and Scale. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: An original length is 6 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 6 × 0.5 = 3.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 3 cm

Worked Example 2

Problem: An original length is 10 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 10 × 1.5 = 15.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 15 cm

Worked Example 3

Problem: An original length is 17 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 17 × 2 = 34.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 34 cm

Worked Example 4

Problem: An original length is 23 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 23 × 2.5 = 57.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 57.5 cm

Worked Example 5

Problem: An original length is 20 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 20 × 3 = 60.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 60 cm

Worked Example 6

Problem: An original length is 14 cm and the scale factor is 0.5. Find the image length.

  1. Multiply the original by 0.5.
  2. 14 × 0.5 = 7.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 7 cm

Worked Example 7

Problem: An original length is 11 cm and the scale factor is 1.5. Find the image length.

  1. Multiply the original by 1.5.
  2. 11 × 1.5 = 16.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 16.5 cm

Worked Example 8

Problem: An original length is 18 cm and the scale factor is 2. Find the image length.

  1. Multiply the original by 2.
  2. 18 × 2 = 36.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 36 cm

Worked Example 9

Problem: An original length is 27 cm and the scale factor is 2.5. Find the image length.

  1. Multiply the original by 2.5.
  2. 27 × 2.5 = 67.5.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 67.5 cm

Worked Example 10

Problem: An original length is 16 cm and the scale factor is 3. Find the image length.

  1. Multiply the original by 3.
  2. 16 × 3 = 48.

Very beginner explanation: A scale factor multiplies every corresponding length by the same number.

Answer: 48 cm

Practice exercise

Create one new Grade 8 problem involving Scale drawings review. Show the important steps, include units when needed, and explain how you checked the answer.

40.12 Real-life similarity

Similar figures have the same shape and proportional corresponding lengths. In this section, the focus is Real-life similarity.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Real-life similarity: Explain Real-life similarity in one simple sentence.

  1. Look at the words in “Real-life similarity”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Real-life similarity is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Real-life similarity: A student says, “I can use Real-life similarity without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Real-life similarity: What is the first step when solving a problem about Real-life similarity?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Real-life similarity: After solving a Real-life similarity problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Real-life similarity: Give one way Real-life similarity could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Real-life similarity can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Real-life similarity: Which representation could help explain Real-life similarity: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Real-life similarity: A student gets an answer for Real-life similarity but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Real-life similarity: Why can estimation help before a detailed Real-life similarity calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Real-life similarity: How can you test whether your rule for Real-life similarity works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Real-life similarity: How would you teach Real-life similarity to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Real-life similarity. Show the important steps, include units when needed, and explain how you checked the answer.

40.13 Applications

Applications is an important Grade 8 concept in Congruence, Similarity, and Scale. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Applications: Explain Applications in one simple sentence.

  1. Look at the words in “Applications”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Applications is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Applications: A student says, “I can use Applications without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Applications: What is the first step when solving a problem about Applications?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Applications: After solving a Applications problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Applications: Give one way Applications could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Applications can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Applications: Which representation could help explain Applications: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Applications: A student gets an answer for Applications but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Applications: Why can estimation help before a detailed Applications calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Applications: How can you test whether your rule for Applications works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Applications: How would you teach Applications to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Applications. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 40 Review Questions and Answers

Q1. What is important to remember about Congruent figures?

Answer: Congruent figures have the same size and shape. In this section, the focus is Congruent figures.

Q2. What is important to remember about Corresponding sides?

Answer: Corresponding sides is an important Grade 8 concept in Congruence, Similarity, and Scale. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, 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. In this section, the focus is Corresponding angles.

Q4. What is important to remember about Similar figures?

Answer: Similar figures have the same shape and proportional corresponding lengths. In this section, the focus is Similar figures.

Q5. What is important to remember about Scale factors?

Answer: A factor (a whole number that divides another number exactly) helps with simplifying fractions and finding common group sizes. In this section, the focus is Scale factors.

Q6. What is important to remember about Similar triangles?

Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Similar triangles.

Q7. What is important to remember about Finding missing sides?

Answer: Finding missing sides is an important Grade 8 concept in Congruence, Similarity, and Scale. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, 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. In this section, the focus is Finding missing angles.

Q9. What is important to remember about Comparing perimeter?

Answer: Perimeter is the total distance around a two-dimensional figure. In this section, the focus is Comparing perimeter.

Q10. What is important to remember about Comparing area?

Answer: Area measures the amount of two-dimensional space inside a figure and uses square units. In this section, the focus is Comparing area.

Q11. What is important to remember about Scale drawings review?

Answer: Scale drawings review is an important Grade 8 concept in Congruence, Similarity, and Scale. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q12. What is important to remember about Real-life similarity?

Answer: Similar figures have the same shape and proportional corresponding lengths. In this section, the focus is Real-life similarity.

Q13. What is important to remember about Applications?

Answer: Applications is an important Grade 8 concept in Congruence, Similarity, and Scale. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.