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Chapter 37: Triangles and Right Triangles

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.

37.1 Classifying triangles by sides

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

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 Classifying triangles by sides. Show the important steps, include units when needed, and explain how you checked the answer.

37.2 Equilateral triangles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Equilateral 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 Equilateral triangles. Show the important steps, include units when needed, and explain how you checked the answer.

37.3 Isosceles triangles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Isosceles 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 Isosceles triangles. Show the important steps, include units when needed, and explain how you checked the answer.

37.4 Scalene triangles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Scalene 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 Scalene triangles. Show the important steps, include units when needed, and explain how you checked the answer.

37.5 Acute triangles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Acute 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 Acute triangles. Show the important steps, include units when needed, and explain how you checked the answer.

37.6 Right triangles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Right 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 Right triangles. Show the important steps, include units when needed, and explain how you checked the answer.

37.7 Obtuse triangles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Obtuse 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 Obtuse triangles. Show the important steps, include units when needed, and explain how you checked the answer.

37.8 Triangle angle sum

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

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 Triangle angle sum. Show the important steps, include units when needed, and explain how you checked the answer.

37.9 Exterior angles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Exterior 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 Exterior angles. Show the important steps, include units when needed, and explain how you checked the answer.

37.10 Missing-angle problems

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

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 Missing-angle problems. Show the important steps, include units when needed, and explain how you checked the answer.

37.11 Triangle inequality

An inequality compares values using symbols such as <, >, ≤, or ≥. In this section, the focus is Triangle inequality.

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: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 2

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 3

Problem: Solve 5x + 7 = 2x + 22.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 4

Problem: Solve 4(x + 2) = 24.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 5

Problem: Solve 7x - 3 = 4x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 6

Problem: Solve 0.5x + 2 = 7.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 10

Worked Example 7

Problem: Solve 2(x-3)+4=10.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 8

Problem: Solve 9 - 2x = 1.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 9

Problem: Solve 3x + 8 = 3x + 8.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: all real numbers

Worked Example 10

Problem: Solve 4x + 1 = 4x + 9.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: no solution

Practice exercise

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

37.12 Right-triangle parts

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Right-triangle parts.

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 Right-triangle parts. Show the important steps, include units when needed, and explain how you checked the answer.

37.13 Real-life triangle problems

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Real-life triangle problems.

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 Real-life triangle problems. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 37 Review Questions and Answers

Q1. What is important to remember about Classifying triangles by sides?

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

Q2. What is important to remember about Equilateral triangles?

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

Q3. What is important to remember about Isosceles triangles?

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

Q4. What is important to remember about Scalene triangles?

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

Q5. What is important to remember about Acute triangles?

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

Q6. What is important to remember about Right triangles?

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

Q7. What is important to remember about Obtuse triangles?

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

Q8. What is important to remember about Triangle angle sum?

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

Q9. What is important to remember about Exterior angles?

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

Q10. What is important to remember about Missing-angle problems?

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

Q11. What is important to remember about Triangle inequality?

Answer: An inequality compares values using symbols such as <, >, ≤, or ≥. In this section, the focus is Triangle inequality.

Q12. What is important to remember about Right-triangle parts?

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

Q13. What is important to remember about Real-life triangle problems?

Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Real-life triangle problems.

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.