Supplementary Angles: Definition, Properties and Examples

Supplementary angles are two angles whose measures add up to 180°. Each angle is called the supplement of the other, so the supplement of an angle x is 180° − x. For example, 110° and 70° are supplementary because 110° + 70° = 180°.

What are supplementary angles?

Two angles are supplementary when their sum is exactly 180°, the angle of a straight line. The two angles do not have to touch or share a vertex. An angle of 130° in one triangle and an angle of 50° in a different figure are still supplementary.

Examples of supplementary pairs:

• 120° and 60°
• 90° and 90°
• 145° and 35°
• 179.5° and 0.5°

Linear pair: adjacent supplementary angles

When two supplementary angles are adjacent (they share a vertex and one arm, and their other arms point in opposite directions), they form a linear pair. Together they make a straight line.

If a ray OC stands on a straight line AB at O, then ∠AOC + ∠COB = 180°. This is the linear pair axiom. Every linear pair is supplementary, but not every supplementary pair is a linear pair, because supplementary angles can be apart.

How to find the supplement of an angle

Subtract the angle from 180°:

Supplement of x = 180° − x

• Supplement of 40° = 180° − 40° = 140°
• Supplement of 125° = 180° − 125° = 55°
• Supplement of 90° = 90°
• Supplement of 63.5° = 116.5°

Only angles between 0° and 180° have a supplement in this sense. An angle of 180° or more has none, because the other angle would be zero or negative.

Supplementary vs complementary angles

Properties of supplementary angles

1. The sum is always 180°. This is the definition.
2. Two acute angles cannot be supplementary. Each is less than 90°, so their sum is less than 180°.
3. Two obtuse angles cannot be supplementary. Each is more than 90°, so their sum is more than 180°.
4. Two right angles are supplementary. 90° + 90° = 180°.
5. Otherwise, one angle is acute and the other obtuse. If one is 70°, the other must be 110°.
6. Supplements of equal angles are equal. If ∠A = ∠B, then 180° − A = 180° − B.
7. The definition covers exactly two angles. Three angles that add to 180°, such as the angles of a triangle, are not called supplementary.
8. Trigonometry: sin(180° − x) = sin x and cos(180° − x) = −cos x. This is why an obtuse angle has a negative cosine, which matters when you use the cosine rule on an obtuse triangle.

Solved examples with algebra

Example 1: x and 2x + 30

Two supplementary angles measure x and (2x + 30)°. Find both.

x + 2x + 30 = 180, so 3x = 150 and x = 50.

The angles are 50° and 130°. Check: 50 + 130 = 180.

Example 2: one angle is four times the other

x + 4x = 180, so 5x = 180 and x = 36. The angles are 36° and 144°.

Example 3: the angles differ by 40°

x + (x + 40) = 180, so 2x = 140 and x = 70. The angles are 70° and 110°.

Example 4: supplement and complement together

The supplement of an angle is three times its complement. Find the angle.

180 − x = 3(90 − x), so 180 − x = 270 − 3x, which gives 2x = 90 and x = 45.

The angle is 45°. Check: supplement 135° = 3 × 45°, its complement.

Supplementary angles with parallel lines

When a transversal cuts two parallel lines, the co-interior angles (also called consecutive interior angles or same-side interior angles) are supplementary. They sit between the parallel lines, on the same side of the transversal. The converse is also true: if co-interior angles add to 180°, the lines are parallel.

Example 5: Two co-interior angles between parallel lines are (3x + 10)° and (2x + 20)°. Find them.

3x + 10 + 2x + 20 = 180, so 5x + 30 = 180, 5x = 150 and x = 30. The angles are 100° and 80°.

Other places supplementary angles appear:

• Adjacent angles of a parallelogram are supplementary.
• Opposite angles of a cyclic quadrilateral are supplementary.
• An interior angle of a polygon and its exterior angle form a linear pair, so they add to 180°.

Common mistakes

• Mixing up 90° and 180°: complementary is 90°, supplementary is 180°.
• Assuming the angles must be adjacent: only a linear pair has to be adjacent.
• Stopping at x: in algebra questions, substitute back to find both angles and check that they add to 180°.
• Calling alternate angles supplementary: with parallel lines, alternate and corresponding angles are equal; co-interior angles are the supplementary ones.

FAQs

What are supplementary angles?

Two angles whose measures add up to 180°. For example, 120° and 60° are supplementary.

Do supplementary angles have to be adjacent?

No. Any two angles that add to 180° are supplementary. When they are adjacent and form a straight line, they are called a linear pair.

Can two acute angles be supplementary?

No. Two acute angles are each less than 90°, so their sum is always less than 180°. Two obtuse angles cannot be supplementary either.

What is the difference between supplementary and complementary angles?

Supplementary angles add up to 180°, complementary angles add up to 90°. The supplement of x is 180° − x, and the complement is 90° − x.

Which angles are supplementary when a transversal cuts parallel lines?

The co-interior angles, the pair of interior angles on the same side of the transversal, add up to 180°.

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