Chapter 45: Angles and Triangles
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Angles and Triangles with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Estimate (a close approximation used to check whether an answer is reasonable)
- Solution (a value or result that satisfies the problem)
- Representation (a way to show mathematics using symbols, diagrams, tables, graphs, or words)
- Reasonableness (whether an answer makes sense in the context of the problem)
How to Learn This Chapter
Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
45.1 Acute angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Acute angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Acute angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Acute angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Acute angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Acute angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Acute angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Acute angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.2 Right angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Right angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Right angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Right angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Right angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Right angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Right angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Right angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.3 Obtuse angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Obtuse angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Obtuse angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Obtuse angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Obtuse angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Obtuse angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Obtuse angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Obtuse angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.4 Straight angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Straight angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Straight angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Straight angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Straight angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Straight angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Straight angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Straight angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.5 Reflex angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Reflex angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Reflex angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Reflex angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Reflex angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Reflex angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Reflex angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Reflex angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.6 Complementary angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Complementary angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find complement of 35°.
- Complementary angles total 90°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Complementary angles .
Answer: 55°
Worked Example 2
Problem: Find complement of 48°.
- Complementary angles total 90°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Complementary angles .
Answer: 42°
Worked Example 3
Problem: Find complement of 67°.
- Complementary angles total 90°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Complementary angles .
Answer: 23°
Worked Example 4
Problem: Find complement of 72°.
- Complementary angles total 90°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Complementary angles .
Answer: 18°
Worked Example 5
Problem: Can 110° have a positive complement?
- Complementary angles total 90°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Complementary angles .
Answer: No
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Complementary angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.7 Supplementary angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Supplementary angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Supplementary angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Supplementary angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Supplementary angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Supplementary angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Supplementary angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Supplementary angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.8 Vertical angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Vertical angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Vertical angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Vertical angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Vertical angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Vertical angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Vertical angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Vertical angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.9 Adjacent angles
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Adjacent angles .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Adjacent angles .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Adjacent angles .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Adjacent angles .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Adjacent angles .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Adjacent angles .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Adjacent angles. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.10 Triangle classifications
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle classifications .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Triangle angles are 35° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle classifications .
Answer: 95°
Worked Example 2
Problem: Triangle angles are 48° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle classifications .
Answer: 82°
Worked Example 3
Problem: Triangle angles are 67° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle classifications .
Answer: 63°
Worked Example 4
Problem: Triangle angles are 72° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle classifications .
Answer: 58°
Worked Example 5
Problem: Triangle angles are 110° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle classifications .
Answer: 20°
Worked Example 6
Problem: A triangle has angles 35° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 35 - 50 = 95.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 95°
Worked Example 7
Problem: A triangle has angles 48° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 48 - 50 = 82.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 82°
Worked Example 8
Problem: A triangle has angles 67° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 67 - 50 = 63.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 63°
Worked Example 9
Problem: A triangle has angles 72° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 72 - 50 = 58.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 58°
Worked Example 10
Problem: A triangle has angles 110° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 110 - 50 = 20.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 20°
Practice Exercise
Create one new question about Triangle classifications. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.11 Triangle angle sum
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle angle sum .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Triangle angles are 35° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle angle sum .
Answer: 95°
Worked Example 2
Problem: Triangle angles are 48° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle angle sum .
Answer: 82°
Worked Example 3
Problem: Triangle angles are 67° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle angle sum .
Answer: 63°
Worked Example 4
Problem: Triangle angles are 72° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle angle sum .
Answer: 58°
Worked Example 5
Problem: Triangle angles are 110° and 50°. Find third.
- Triangle angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle angle sum .
Answer: 20°
Worked Example 6
Problem: A triangle has angles 35° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 35 - 50 = 95.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 95°
Worked Example 7
Problem: A triangle has angles 48° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 48 - 50 = 82.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 82°
Worked Example 8
Problem: A triangle has angles 67° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 67 - 50 = 63.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 63°
Worked Example 9
Problem: A triangle has angles 72° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 72 - 50 = 58.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 58°
Worked Example 10
Problem: A triangle has angles 110° and 50°. Find the third angle.
- Triangle angles total 180°.
- 180 - 110 - 50 = 20.
Very beginner explanation: Subtract the known angles from 180°.
Answer: 20°
Practice Exercise
Create one new question about Triangle angle sum. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.12 Exterior angles introduction
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Exterior angles introduction .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Exterior angles introduction .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Exterior angles introduction .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Exterior angles introduction .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Exterior angles introduction .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Exterior angles introduction .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Exterior angles introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
45.13 Unknown-angle problems
An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Unknown-angle problems .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find supplementary angle to 35°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Unknown-angle problems .
Answer: 145°
Worked Example 2
Problem: Find supplementary angle to 48°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Unknown-angle problems .
Answer: 132°
Worked Example 3
Problem: Find supplementary angle to 67°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Unknown-angle problems .
Answer: 113°
Worked Example 4
Problem: Find supplementary angle to 72°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Unknown-angle problems .
Answer: 108°
Worked Example 5
Problem: Find supplementary angle to 110°.
- Supplementary angles total 180°.
Very beginner explanation: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Unknown-angle problems .
Answer: 70°
Worked Example 6
Problem: Find the supplementary angle to 35°.
- Supplementary angles total 180°.
- 180 - 35 = 145.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 145°
Worked Example 7
Problem: Find the supplementary angle to 48°.
- Supplementary angles total 180°.
- 180 - 48 = 132.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 132°
Worked Example 8
Problem: Find the supplementary angle to 67°.
- Supplementary angles total 180°.
- 180 - 67 = 113.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 113°
Worked Example 9
Problem: Find the supplementary angle to 72°.
- Supplementary angles total 180°.
- 180 - 72 = 108.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 108°
Worked Example 10
Problem: Find the supplementary angle to 110°.
- Supplementary angles total 180°.
- 180 - 110 = 70.
Very beginner explanation: Angles on a straight line total 180°.
Answer: 70°
Practice Exercise
Create one new question about Unknown-angle problems. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
Chapter Notes, Practice, and Common Mistakes
Chapter Notes
- Read the complete question before choosing an operation, formula, graph, or model.
- Write technical words together with their meaning until you are comfortable using them.
- Show all important steps so another student can follow your reasoning.
- Keep units, labels, signs, axes, variables, and mathematical symbols clear.
- Estimate before or after calculating when an estimate can help check reasonableness.
- For real-life problems, explain what the final number means in the situation.
Extra Practice
- Create and solve a new question about Acute angles . Show your reasoning and check your answer.
- Create and solve a new question about Right angles . Show your reasoning and check your answer.
- Create and solve a new question about Obtuse angles . Show your reasoning and check your answer.
- Create and solve a new question about Straight angles . Show your reasoning and check your answer.
- Create and solve a new question about Reflex angles . Show your reasoning and check your answer.
- Create and solve a new question about Complementary angles . Show your reasoning and check your answer.
- Create and solve a new question about Supplementary angles . Show your reasoning and check your answer.
- Create and solve a new question about Vertical angles . Show your reasoning and check your answer.
- Create and solve a new question about Adjacent angles . Show your reasoning and check your answer.
- Create and solve a new question about Triangle classifications . Show your reasoning and check your answer.
- Create and solve a new question about Triangle angle sum . Show your reasoning and check your answer.
- Create and solve a new question about Exterior angles introduction . Show your reasoning and check your answer.
Common Mistakes
- Assuming a diagram is drawn to scale.
- Confusing radius and diameter.
- Using square units for volume or cubic units for area.
- Applying a scale factor to only one dimension.
- Forgetting the circular bases in cylinder surface area.
30 Review Questions and Answers
Q1. What is important to remember about Acute angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Acute angles.
Q2. What is important to remember about Right angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Right angles.
Q3. What is important to remember about Obtuse angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Obtuse angles.
Q4. What is important to remember about Straight angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Straight angles.
Q5. What is important to remember about Reflex angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Reflex angles.
Q6. What is important to remember about Complementary angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Complementary angles.
Q7. What is important to remember about Supplementary angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Supplementary angles.
Q8. What is important to remember about Vertical angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Vertical angles.
Q9. What is important to remember about Adjacent angles?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Adjacent angles.
Q10. What is important to remember about Triangle classifications?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Triangle classifications.
Q11. 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. This topic focuses on Triangle angle sum.
Q12. What is important to remember about Exterior angles introduction?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Exterior angles introduction.
Q13. What is important to remember about Unknown-angle problems?
Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. This topic focuses on Unknown-angle 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 final answer is reasonable before accepting it.
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 of a calculation 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 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, check copied values, signs, units, formulas, and calculations.
Q22. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q23. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q24. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q25. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.
Q26. How should you study this chapter?
Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.
Q27. When is a calculator useful?
Answer: A calculator is useful for lengthy arithmetic after you understand the mathematical setup and can estimate the expected size of the answer.
Q28. Why compare more than one strategy?
Answer: Different strategies can make a problem easier and provide a way to verify the result.
Q29. What is a mathematical model?
Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.
Q30. Why should assumptions be stated?
Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.