Complementary
Two angles that add to 90°.
Learning Objectives
- use angles on a straight line to total 180°;
- use angles around a point to total 360°;
- solve missing-angle equations;
- identify complementary and supplementary angles;
- explain each calculation using an angle rule.
Introduction
Angles describe turns. When several angles fit along a straight path or meet around one point, the complete turn gives us a total we can use.
Angles on a straight line add up to 180°.
Key Notes
Angles around a point add up to 360°.
Angles around a point
Three angles around a point are 85°, 120° and x.
x = 360° − 85° − 120° = 155°.
Complementary and Supplementary
Supplementary
Two angles that add to 180°.
Algebra in Angle Problems
Algebra in angle problems
Adjacent angles on a straight line are (3x+10)° and (2x−5)°.
(3x+10)+(2x−5)=180
5x+5=180, so x=35.
The angles are 115° and 65°.
Try It: Angles on a Straight Line
70° + 110° = 180°
| First angle | Second angle | Total |
|---|---|---|
| 70° | 110° | 180° |
🧠 Let’s Think Together
A straight line is split into angles of 47° and x. The two turns together make 180°, so x is the amount still needed: 180° − 47° = 133°.
First identify the complete turn shown by the diagram; only then choose the calculation.
Worked Examples
Example 1: Angles around a point
1
Add the known angles: 90° + 75° + 110° = 275°.
2
Subtract from the full turn: x = 360° − 275° = 85°.
Example 2: Algebra on a straight line
(3x + 10)° + (2x − 5)° = 180°, so 5x + 5 = 180 and x = 35. The angles are 115° and 65°.
Practice Questions
Foundation- Find x if x+47°=180°.
- Angles around a point are 90°, 75°, 110° and x. Find x.
- Find the complement of 38°.
- Two supplementary angles are in the ratio 2:3. Find them.
- Angles on a straight line are (4x+8)° and (2x+16)°. Find x and both angles.
- Four angles around a point are x, 2x, 3x and 4x. Find all four angles.
✅ Check Your Work
Show practice answers
- 133°.
- 85°.
- 52°.
- 72° and 108°.
- 6x+24=180, x=26. Angles: 112° and 68°.
- 10x=360, x=36. Angles: 36°, 72°, 108°, 144°.
Common Mistakes
- Using 180° for angles around a point instead of 360°.
- Subtracting each known angle separately without first finding their total.
- Giving a value for x but not calculating the requested angle.
Exam Focus
Write the angle rule beside your calculation. Check whether the diagram shows a straight line, a right angle or a full turn, and include degrees in the final answer.
Summary
- Straight line: 180°.
- Around a point: 360°.
- Complementary: 90°.
- Supplementary: 180°.