What Are Complementary Angles? Definition, Examples & Practice Problems

Complementary angles are two angles whose measures add up to exactly 90 degrees — a right angle. They don’t need to touch or sit next to each other; as long as the two measures total 90°, the pair counts as complementary, and each individual angle must be acute (under 90°).

Still mixing up complementary with supplementary angles? This guide walks through the definition, real-world examples, and practice problems so the difference sticks.

What Is the Definition of Complementary Angles?

Complementary angles are two angles whose measures add up to 90 degrees (a right angle).

Key formula:

∠A + ∠B = 90°

Important: Complementary angles do NOT need to be next to each other. They can be anywhere – as long as their sum equals 90°, they’re complementary!

What Do Complementary Angles Look Like?

Example 1: Adjacent Complementary Angles

[IMAGE: Two angles sharing a side, measuring 30° and 60°, forming a right angle together]

In the diagram above:

• ∠A = 30°
• ∠B = 60°
• 30° + 60° = 90° ✓
• These angles are both adjacent AND complementary

Example 2: Non-Adjacent Complementary Angles

[IMAGE: Two separate angles in different locations, one 25° and one 65°]

These angles are complementary but NOT adjacent because:

• ∠C = 25°
• ∠D = 65°
• 25° + 65° = 90° ✓
• They’re in different locations (not touching)

Where Do You See Complementary Angles in Real Life?

Example 1: Right Triangle

In any right triangle, the two acute angles are always complementary. For example, a 30-60-90 triangle has two complementary angles: 30° + 60° = 90°.

Example 2: Open Book

When you open a book partway, the angle between the cover and the first page plus the angle between the last page and the back cover often add up to 90°.

Example 3: L-Shaped Corner

An L-shaped desk or countertop has a 90° corner. Any two angles that fit together to fill that corner are complementary — the same corner geometry shows up in the L-shaped kitchen counters common in Côte-Saint-Luc bungalows.

Example 4: Ramp Angles

If a wheelchair ramp has a 15° incline from the ground, the angle between the ramp surface and a vertical wall is 75° (because 15° + 75° = 90°) — a calculation contractors run when adding an accessibility ramp to a Villeray duplex entrance.

What Are the Key Properties of Complementary Angles?

✅ Complementary angles always sum to 90°
✅ Can be adjacent or non-adjacent
✅ Both angles must be acute (each less than 90°)
✅ Can be equal (both 45°) or different sizes
❌ Do NOT confuse with supplementary (180°)

How Can You Remember Complementary vs. Supplementary?

💡 How to Remember:
Complementary = Corner (90° right angle)
Supplementary = Straight line (180°)

Alphabetically: C comes before S, and 90 comes before 180!

How Do Complementary Angles Compare to Supplementary and Adjacent Angles?

How to Find Complementary Angles

Formula: If you know one angle, subtract it from 90° to find its complement.

Complement of angle A = 90° – A

Examples:

• Complement of 30° = 90° – 30° = 60°
• Complement of 45° = 90° – 45° = 45°
• Complement of 72° = 90° – 72° = 18°
• Complement of 15° = 90° – 15° = 75°

What Mistakes Do Students Make With Complementary Angles?

❌ Mistake #1: Confusing complementary with supplementary

WRONG: « Complementary angles add to 180° »
CORRECT: Complementary = 90°, Supplementary = 180°

❌ Mistake #2: Thinking complementary angles must be adjacent

WRONG: « Complementary angles must touch each other »
CORRECT: They can be anywhere – only the sum matters (90°)

❌ Mistake #3: Using angles larger than 90°

WRONG: « 95° and -5° are complementary »
CORRECT: Both angles must be positive and less than 90° (acute)

❌ Mistake #4: Forgetting to subtract from 90°

WRONG: Finding complement of 35° → 180° – 35° = 145°
CORRECT: Complement of 35° → 90° – 35° = 55°

Practice Problems

Basic Problems

1. Problem 1: Find the complement of 40°.
   Solution: 90° – 40° = 50°
   Check: 40° + 50° = 90° ✓
2. Problem 2: Are angles measuring 55° and 35° complementary?
   Solution: 55° + 35° = 90° → YES ✓
3. Problem 3: Two complementary angles are equal. What’s the measure of each angle?
   Solution: x + x = 90° → 2x = 90° → x = 45°
   Check: 45° + 45° = 90° ✓

Intermediate Problems

4. Problem 4: One angle measures 28°. What’s its complement?
   Solution: 90° – 28° = 62°
   Check: 28° + 62° = 90° ✓
5. Problem 5: Two complementary angles are in the ratio 2:3. Find each angle.
   Solution:
   Let the angles be 2x and 3x
   2x + 3x = 90°
   5x = 90°
   x = 18°
   First angle: 2(18°) = 36°
   Second angle: 3(18°) = 54°
   Check: 36° + 54° = 90° ✓
6. Problem 6: In a right triangle, one acute angle is 32°. What’s the other acute angle?
   Solution: The two acute angles in a right triangle are always complementary.
   90° – 32° = 58°

Advanced Problems

7. Problem 7: Two complementary angles have measures (2x + 10)° and (3x – 5)°. Find the value of x and both angle measures.
   Solution:
   (2x + 10) + (3x – 5) = 90
   5x + 5 = 90
   5x = 85
   x = 17
   First angle: 2(17) + 10 = 44°
   Second angle: 3(17) – 5 = 46°
   Check: 44° + 46° = 90° ✓
8. Problem 8: The larger of two complementary angles is 18° more than twice the smaller angle. Find both angles.
   Solution:
   Let the smaller angle = x
   Larger angle = 2x + 18
   x + (2x + 18) = 90
   3x + 18 = 90
   3x = 72
   x = 24
   Smaller angle: 24°
   Larger angle: 2(24) + 18 = 66°
   Check: 24° + 66° = 90° ✓

Challenge Problem

9. Problem 9: Three angles form a right angle together. The first angle is x°, the second is (x + 10)°, and the third is (2x – 10)°. Are any two of these angles complementary to each other?
   Solution:
   First, find x: x + (x + 10) + (2x – 10) = 90
   4x = 90
   x = 22.5°
   First angle: 22.5°
   Second angle: 32.5°
   Third angle: 35°
   Check pairs: 22.5° + 32.5° = 55° ✗, 22.5° + 35° = 57.5° ✗, 32.5° + 35° = 67.5° ✗
   Answer: No pairs are complementary (they only sum to 90° as a group of three)

How Do Complementary Angles Show Up in Word Problems?

10. Problem 10: A ladder leans against a wall at a 35° angle from the ground. What’s the angle between the ladder and the wall?
    Solution: The ground and wall form a 90° angle, so:
    90° – 35° = 55°
11. Problem 11: A carpenter cuts a board at a 64° angle. What angle does he need to cut to make a perfect 90° corner joint?
    Solution: 90° – 64° = 26°

Why Do Complementary Angles Matter in Math?

Understanding complementary angles is essential for:

• Right triangles: The two acute angles are always complementary
• Trigonometry: sin(x) = cos(90° – x) – this relationship is based on complementary angles
• Geometry proofs: Many theorems involve complementary angle relationships
• Real-world applications: Construction, engineering, and design
• Quebec curriculum: Essential for Secondary 2-3 and examens ministériels

What Other Angle Types Should You Learn Next?

Now that you understand complementary angles, learn about:

• Supplementary Angles – angles that add to 180°
• Adjacent Angles – angles that share a vertex and side
• Vertical Angles – opposite angles formed by intersecting lines
• Linear Pairs – adjacent supplementary angles

Summary

Key Takeaways:

• Complementary angles always add up to 90°
• They don’t need to be adjacent or next to each other
• To find a complement: subtract from 90°
• Both angles must be acute (less than 90°)
• Remember: C for Corner (90°), S for Straight line (180°)

Frequently Asked Questions (FAQ)