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Supplement

Supplement of an Angle

For any angle A between 0° and 180°, the supplement of A is 180° – A.

 

 

 

See also

Supplementary angles, complement of an angle

Key Formula

Supplement of A=180°−A\text{Supplement of } A = 180° - A
Where:
  • AA = The measure of the original angle, where 0° < A < 180°

Worked Example

Problem: Find the supplement of a 55° angle.
Step 1: Write the supplement formula.
Supplement=180°−A\text{Supplement} = 180° - A
Step 2: Substitute A = 55°.
180°−55°=125°180° - 55° = 125°
Answer: The supplement of 55° is 125°. Notice that 55° + 125° = 180°, confirming the two angles are supplementary.

Another Example

Problem: Two angles are supplementary. One angle is three times the other. Find both angles.
Step 1: Let the smaller angle be x. Then the larger angle is 3x.
x+3x=180°x + 3x = 180°
Step 2: Combine like terms and solve.
4x=180°  ⟹  x=45°4x = 180° \implies x = 45°
Step 3: Find the larger angle.
3x=3(45°)=135°3x = 3(45°) = 135°
Answer: The two angles are 45° and 135°. Each is the supplement of the other.

Frequently Asked Questions

Can an angle greater than 180° have a supplement?
By the standard definition, no. The supplement is defined only for angles strictly between 0° and 180°. If A = 180°, the supplement would be 0°, and if A > 180°, the formula gives a negative result, which does not represent a valid angle measure in basic geometry.
What is the supplement of a 90° angle?
The supplement of 90° is 180° − 90° = 90°. This means a right angle is its own supplement. It also means two right angles placed together form a straight angle (180°).

Supplement (180°) vs. Complement (90°)

The supplement of an angle is what you add to reach 180°, while the complement is what you add to reach 90°. For example, the supplement of 60° is 120°, but the complement of 60° is 30°. Complements exist only for angles less than 90°, whereas supplements exist for angles less than 180°. A common memory trick: "C" comes before "S" in the alphabet, and 90 comes before 180.

Why It Matters

Supplementary angles appear whenever two angles form a straight line, such as a linear pair created by two intersecting lines. In triangle geometry, knowing that the interior angles sum to 180° means each angle is the supplement of the sum of the other two. Understanding supplements is also essential for working with co-interior (same-side interior) angles formed by a transversal crossing parallel lines, which are always supplementary.

Common Mistakes

Mistake: Confusing supplement with complement — subtracting from 90° instead of 180°.
Correction: Supplement always involves 180° (think "s" for "straight angle"). Complement involves 90° (think "c" for "corner" or right angle).
Mistake: Claiming that an obtuse angle (greater than 90°) has no supplement.
Correction: Any angle between 0° and 180° has a supplement. For instance, the supplement of 140° is 40°. The supplement of an obtuse angle is simply an acute angle.

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