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Supplementary Angles

Supplementary Angles

Two angles that add up to 180°.

 

Two angles labeled 1 and 2 formed by rays on a straight line, illustrating that angle 1 + angle 2 = 180°.

 

 

See also

Supplement of an angle, complementary angles, straight angle, measure of an angle

Key Formula

∠A+∠B=180°\angle A + \angle B = 180°
Where:
  • ∠A\angle A = The measure of the first angle in degrees
  • ∠B\angle B = The measure of the second angle in degrees

Worked Example

Problem: One angle measures 65°. Find its supplementary angle.
Step 1: Write the supplementary angle formula.
∠A+∠B=180°\angle A + \angle B = 180°
Step 2: Substitute the known angle into the formula.
65°+∠B=180°65° + \angle B = 180°
Step 3: Solve for the unknown angle by subtracting 65° from both sides.
∠B=180°−65°=115°\angle B = 180° - 65° = 115°
Step 4: Verify: check that the two angles sum to 180°.
65°+115°=180°✓65° + 115° = 180° \checkmark
Answer: The supplementary angle is 115°.

Another Example

This example introduces a ratio constraint instead of giving one angle directly, requiring students to set up and solve an algebraic equation.

Problem: Two supplementary angles are in the ratio 2 : 7. Find both angles.
Step 1: Let the two angles be 2x2x and 7x7x, where xx is a common multiplier.
2x+7x=180°2x + 7x = 180°
Step 2: Combine like terms.
9x=180°9x = 180°
Step 3: Solve for xx.
x=180°9=20°x = \frac{180°}{9} = 20°
Step 4: Find each angle by substituting back.
2x=2(20°)=40°and7x=7(20°)=140°2x = 2(20°) = 40° \quad\text{and}\quad 7x = 7(20°) = 140°
Step 5: Verify the sum.
40°+140°=180°✓40° + 140° = 180° \checkmark
Answer: The two angles are 40° and 140°.

Frequently Asked Questions

What is the difference between supplementary and complementary angles?
Supplementary angles add up to 180°, while complementary angles add up to 90°. A quick way to remember: the 's' in supplementary stands for 'straight' (a straight angle is 180°), and the 'c' in complementary stands for 'corner' (a right-angle corner is 90°).
Do supplementary angles have to be next to each other?
No. Two angles are supplementary as long as their measures sum to 180°, regardless of their position. When they are placed adjacent so they share a common side, they form a straight line — but they can also be in completely different locations and still be supplementary.
Can two acute angles be supplementary?
No. An acute angle is less than 90°, so two acute angles would add up to less than 180°. At least one of the two supplementary angles must be obtuse (greater than 90°), unless both are exactly 90°.

Supplementary Angles vs. Complementary Angles

Supplementary AnglesComplementary Angles
DefinitionTwo angles that add up to 180°Two angles that add up to 90°
Formula∠A + ∠B = 180°∠A + ∠B = 90°
Related angle typeStraight angle (180°)Right angle (90°)
Example pair60° and 120°60° and 30°
Can both be acute?NoYes — both must be acute
Memory aid'S' for Straight (180°)'C' for Corner (90°)

Why It Matters

Supplementary angles appear constantly in geometry — whenever two angles sit on a straight line (a linear pair), they are supplementary. You rely on this relationship to find unknown angles in triangles, parallel-line problems, and polygon calculations. Standardized tests like the SAT and ACT regularly include questions that require recognizing supplementary angle pairs to solve for missing values.

Common Mistakes

Mistake: Confusing supplementary (180°) with complementary (90°).
Correction: Remember: 'S' is for Straight line = 180°, and 'C' is for Corner (right angle) = 90°. Supplementary is the larger sum.
Mistake: Assuming supplementary angles must be adjacent or physically touching.
Correction: Two angles are supplementary based solely on their measures summing to 180°. They do not need to share a vertex or a side. Adjacent supplementary angles (a linear pair) are just one special case.

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