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Complementary Angles

Complementary Angles

Two acute angles that add up to 90°. For example, 40° and 50° are complementary. In the diagram below, angles 1 and 2 are complementary.

 

Two rays forming a right angle at a corner, with angle 1 above and angle 2 below, together making 90°.

 

 

See also

Complement of an angle, supplementary angles, measure of an angle

Key Formula

∠A+∠B=90°\angle A + \angle B = 90°
Where:
  • ∠A\angle A = The measure of the first angle
  • ∠B\angle B = The measure of the second angle

Worked Example

Problem: One angle measures 35°. What is the measure of its complement?
Step 1: Write the complementary angle formula.
∠A+∠B=90°\angle A + \angle B = 90°
Step 2: Substitute the known angle into the formula.
35°+∠B=90°35° + \angle B = 90°
Step 3: Solve for the unknown angle by subtracting 35° from both sides.
∠B=90°−35°=55°\angle B = 90° - 35° = 55°
Step 4: Verify: check that the two angles sum to 90°.
35°+55°=90°✓35° + 55° = 90° \checkmark
Answer: The complement of 35° is 55°.

Another Example

This example uses a ratio instead of a single given angle, requiring students to set up and solve an algebraic equation — a common variation in homework and test problems.

Problem: Two complementary angles are in the ratio 2 : 3. Find the measure of each angle.
Step 1: Let the two angles be 2x and 3x, where x is a common multiplier.
∠A=2x,∠B=3x\angle A = 2x, \quad \angle B = 3x
Step 2: Since the angles are complementary, set their sum equal to 90°.
2x+3x=90°2x + 3x = 90°
Step 3: Combine like terms and solve for x.
5x=90°  ⟹  x=18°5x = 90° \implies x = 18°
Step 4: Find each angle by substituting x back in.
∠A=2(18°)=36°,∠B=3(18°)=54°\angle A = 2(18°) = 36°, \quad \angle B = 3(18°) = 54°
Step 5: Verify the result.
36°+54°=90°✓36° + 54° = 90° \checkmark
Answer: The two angles measure 36° and 54°.

Frequently Asked Questions

What is the difference between complementary and supplementary angles?
Complementary angles add up to 90°, while supplementary angles add up to 180°. A quick memory trick: "C" comes before "S" in the alphabet, and 90 comes before 180. Both complementary angles must be acute (less than 90°), whereas supplementary angles can include obtuse angles.
Do complementary angles have to be next to each other?
No. Two angles are complementary as long as their measures sum to 90°. They do not need to be adjacent (sharing a side) or even in the same diagram. For example, an angle in one triangle and an angle in a completely different figure can be complementary if their measures add to 90°.
Can a right angle or an obtuse angle have a complement?
No. Both complementary angles must be acute, meaning each must measure less than 90°. A 90° angle would need a 0° complement, which is not a valid angle in standard geometry. An obtuse angle (greater than 90°) would require a negative complement, which does not exist.

Complementary Angles vs. Supplementary Angles

Complementary AnglesSupplementary Angles
DefinitionTwo angles whose measures sum to 90°Two angles whose measures sum to 180°
Formula∠A + ∠B = 90°∠A + ∠B = 180°
Angle typesBoth angles must be acute (< 90°)One or both can be obtuse; both can be right (90° each)
Example pair25° and 65°110° and 70°
Memory aid"C" for Corner (a right-angle corner is 90°)"S" for Straight (a straight line is 180°)

Why It Matters

Complementary angles appear frequently in right triangles: the two non-right angles are always complementary because the three interior angles of any triangle sum to 180°, and the right angle already accounts for 90°. This relationship is central to trigonometry, where the sine of an angle equals the cosine of its complement (the origin of the name "co-sine"). You will also encounter complementary angles in geometry proofs, coordinate geometry, and standardized tests like the SAT and ACT.

Common Mistakes

Mistake: Confusing complementary (sum = 90°) with supplementary (sum = 180°).
Correction: Remember: Complementary → Corner (90°), Supplementary → Straight line (180°). The letter "C" comes before "S", just as 90 comes before 180.
Mistake: Assuming complementary angles must be adjacent or part of the same figure.
Correction: Complementary is a relationship based solely on angle measures, not position. Two angles anywhere can be complementary if their measures add to 90°.

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