Identifying Complementary Angles- Practice Problems and Solutions
What Are Complementary Angles?
Complementary angles are two angles that add up to exactly 90 degrees. That's it. No tricks, no hidden meaning. If two angles sum to 90°, they're complementary.
These angles don't need to be adjacent or touching. They just need to be a pair that equals a right angle when combined.
Key Facts to Memorize
- Complementary = 90° total
- Supplementary = 180° total
- Vertical angles = equal to each other
Mix these up and you'll lose points. It's that simple.
How to Identify Complementary Angles
Here's the straightforward process:
- Find the given angle measurement
- Subtract it from 90°
- The result is the complementary angle
Example: If one angle is 35°, the complementary angle is 90° - 35° = 55°.
Check your work: 35° + 55° = 90°. Done.
Complementary vs. Supplementary: The Quick Comparison
| Type | Sum | Example |
|---|---|---|
| Complementary | 90° | 30° + 60° |
| Supplementary | 180° | 110° + 70° |
| Linear Pair | 180° + adjacent | 120° + 60° on a straight line |
Don't confuse complementary with supplementary. Students do it constantly on tests. 90° is complementary. 180° is supplementary.
Practice Problems
Try these before checking the solutions. No peeking.
Problem 1
Angle A measures 32°. What is the measurement of Angle B if they are complementary?
Problem 2
If the ratio of two complementary angles is 2:7, find both angle measurements.
Problem 3
An angle measures (3x + 15)° and its complement measures (2x + 25)°. Find the value of x.
Problem 4
One angle is 12° more than half its complement. What are both angles?
Problem 5
Two complementary angles differ by 14°. Find both angles.
Solutions
Solution 1
90° - 32° = 58°
Verification: 32° + 58° = 90° ✓
Solution 2
Let the angles be 2x and 7x.
2x + 7x = 90°
9x = 90°
x = 10°
First angle: 2(10) = 20°
Second angle: 7(10) = 70°
Check: 20° + 70° = 90° ✓
Solution 3
Set up the equation:
(3x + 15) + (2x + 25) = 90°
5x + 40 = 90°
5x = 50°
x = 10°
First angle: 3(10) + 15 = 45°
Second angle: 2(10) + 25 = 45°
Interesting result — they're equal! 45° + 45° = 90° ✓
Solution 4
Let the complement = x°
The angle = (x/2) + 12°
Set up: x + (x/2 + 12) = 90°
1.5x + 12 = 90°
1.5x = 78°
x = 52°
Complement: 52°
The angle: 52/2 + 12 = 38°
Check: 38° + 52° = 90° ✓
Solution 5
Let the smaller angle = x°
The larger angle = x + 14°
x + (x + 14) = 90°
2x + 14 = 90°
2x = 76°
x = 38°
Smaller angle: 38°
Larger angle: 38 + 14 = 52°
Check: 38° + 52° = 90° ✓
Common Mistakes to Avoid
- Confusing with supplementary — Some students automatically use 180° when they should use 90°. Read the problem twice.
- Algebra errors — When angles are expressed as variables, solve the equation step by step. Don't skip the setup.
- Forgetting to check — Always add your two answers to verify they equal 90°. It takes 3 seconds and catches mistakes.
- Assuming angles are equal — Complementary angles don't have to be congruent. Only when each is 45° are they both equal.
Quick Reference Table
| Angle A | Complement (Angle B) |
|---|---|
| 10° | 80° |
| 20° | 70° |
| 30° | 60° |
| 40° | 50° |
| 45° | 45° |
| 55° | 35° |
| 75° | 15° |
Notice 45° + 45° is the only pair where both angles are identical. That's worth remembering.
How to Get Faster at These Problems
Practice with number bonds to 90. Think: 30 and 60, 15 and 75, 40 and 50. The more familiar you are with pairs that sum to 90, the faster you solve these on tests.
When you see a ratio problem, immediately set up 2x + 7x = 90. When you see "differ by X," immediately set up x + (x + X) = 90. These are patterns. Learn them.