Two angles are supplementary to each other when their angle measures add up to exactly 180 degrees, or a straight angle. So naturally, for example, a 60° angle and a 120° angle are supplementary because (60 + 120 = 180). Understanding supplementary angles is essential in geometry because the relationship appears in triangle calculations, parallel lines, polygons, angle-chasing problems, and many real-world applications Simple, but easy to overlook. That alone is useful..
What Does Supplementary Mean?
The word supplementary comes from a Latin term meaning “to complete.” In geometry, two angles complete one another when their combined measure forms a straight angle Still holds up..
If angles (A) and (B) are supplementary, they can be written as:
[ A + B = 180^\circ ]
This relationship works regardless of whether the angles are next to each other. They may share a vertex, be separated, overlap, or belong to entirely different figures. The only requirement is that their measures total 180° And that's really what it comes down to..
For instance:
- 30° and 150°
- 45° and 135°
- 75° and 105°
- 80° and 100°
- 90° and 90°
All of these pairs are supplementary.
Which Angles Are Supplementary to Each Other?
Any two angles whose measures add up to 180° are supplementary. In real terms, there is no single fixed pair that always works. Instead, the measure of the second angle depends on the measure of the first.
Common Supplementary Angle Pairs
| First angle | Supplementary angle |
|---|---|
| 10° | 170° |
| 20° | 160° |
| 25° | 155° |
| 30° | 150° |
| 35° | 145° |
| 40° | 140° |
| 45° | 135° |
| 50° | 130° |
| 55° | 125° |
| 60° | 120° |
| 65° | 115° |
| 70° | 110° |
| 75° | 105° |
| 80° | 100° |
| 85° | 95° |
| 90° | 90° |
People argue about this. Here's where I land on it.
The pairs do not have to have matching “special” values. Here's one way to look at it: 37° and 143° are supplementary even though neither angle is a commonly memorized value.
Supplementary Angles and Linear Pairs
A linear pair is one of the most familiar examples of supplementary angles. It consists of two angles that:
- Share the same vertex.
- Have adjacent sides that form a straight line.
- Do not overlap.
- Add up to 180°.
Imagine a straight line passing through a point (O), with one ray extending to the left and another extending to the right. Because of that, a third ray divides the space above the line into two angles. Those two angles form a linear pair, so their measures must total 180°.
If one angle measures 48°, the other must measure:
[ 180^\circ - 48^\circ = 132^\circ ]
Every linear pair is therefore a pair of supplementary angles. That said, not every pair of supplementary angles forms a linear pair. Supplementary angles only need to total 180°; a linear pair has the additional requirement that the angles are adjacent and their outer sides form a straight line Took long enough..
How to Find an Angle That Is Supplementary to Another
To find the angle supplementary to a given angle, subtract the known measure from 180°.
Step 1: Identify the known angle
Suppose the known angle measures 74°.
Step 2: Use the supplementary-angle equation
[ \text{Unknown angle} = 180^\circ - 74^\circ ]
Step 3: Calculate the result
[ \text{Unknown angle} = 106^\circ ]
So, 74° and 106° are supplementary to each other.
A helpful way to remember the rule is:
What is missing from 180°?
If the known angle is 65°, then (180^\circ - 65^\circ = 115^\circ). If it is 110°, then the supplementary angle is 70°.
Worked Examples
Example 1: Finding a Missing Angle
Two adjacent angles form a straight line. Which means one angle measures 37°. Find the measure of the other It's one of those things that adds up..
Because the angles are supplementary:
[ x + 37^\circ = 180^\circ ]
Subtract 37° from both sides:
[ x = 143^\circ ]
The missing angle is 143°.
Example 2: Solving an Equation
If two supplementary angles are represented by (3x + 10^\circ)
