What Are Supplementary Angles?

Supplementary angles are two angles whose measures add up to 180°. Each angle is called the supplement of the other.

Example: 120° and 60° are supplementary, because 120 + 60 = 180.

More pairs:

• 90° and 90°
• 150° and 30°
• 45° and 135°
• 100° and 80°
• 1° and 179°

Supplementary vs. complementary — the classic mix-up:

How to remember which is which:

• C comes before S in the alphabet, and 90 comes before 180. So Complementary = 90°, Supplementary = 180°.
• Or: C for Corner (a right angle), S for Straight (a straight line).

Finding a missing angle: subtract from 180.

If one angle is 65°, its supplement is 180 − 65 = 115°.

In algebra problems, set the two expressions equal to 180. If the angles are x and 3x, then x + 3x = 180, so 4x = 180 and x = 45° — giving angles of 45° and 135°.

Adjacent vs. non-adjacent supplementary angles:

• Adjacent supplementary angles share a vertex and a side, and together form a straight line. This pair is called a linear pair, and every linear pair is supplementary.
• Non-adjacent supplementary angles can sit anywhere — in different places on a page, or in different shapes entirely. They're still supplementary as long as they total 180°.

So every linear pair is supplementary, but not every supplementary pair is a linear pair.

Key properties:

• Supplementary angles always come in pairs — you can't call three angles supplementary even if they sum to 180°.
• If both angles are equal, each must be 90°.
• One angle is acute and the other obtuse, unless both are exactly 90°.
• Two obtuse angles can never be supplementary (they'd exceed 180°), and neither can two acute angles (they'd fall short).
• Angles supplementary to the same angle are equal to each other.

Where they show up in geometry:

• Angles on a straight line always sum to 180°.
• Co-interior (same-side interior) angles formed when a transversal crosses two parallel lines are supplementary.
• Consecutive angles in a parallelogram are supplementary.
• Opposite angles in a cyclic quadrilateral (one inscribed in a circle) are supplementary.
• The angles of a triangle sum to 180°, so any two of them are supplementary to the third's exterior angle.

Real-world examples: the two angles formed where a ladder leans against a wall and meets the ground, the hands of a clock at 6:00 (180° apart, forming a straight line), and the angles either side of a hinge opened flat.