What are Supplementary and Complementary angles?
Supplementary angles and complementary angles are defined with respect to a pair of angles.
If a right angle, that is 90°, is split into two adjacent angles, then these two angles are known as complementary angles. A pair of complementary angles always adds up to 90°.
Similarly, if a straight line, that is 180°, is split into two adjacent angles, then these two angles are known as supplementary angles. A supplementary pair of angles adds up to 180°.
Supplementary angles and Complementary angles formula
There are two main formulas that are applied to a pair of angles to understand whether they add up to 90° or 180°.
- Two angles ∠A and ∠B are complementary pair of angles if:
- ∠A + ∠B = 90°
- Two angles ∠C and ∠D are supplementary pair of angles if:
- ∠C + ∠D = 180°
We will discuss these formulas in detail in the next section.
Note: Angles can be measured in degrees or in radians. Here we will use degrees to measure angles.
How do we use the Supplementary and Complementary angles formula?
If the sum of the measure of two angles is 90°, then they are complementary angles. In other words, if two angles add up to form a right angle, then they are called complementary angles. It is like the two angles are complementing each other to form a right angle.
Let’s consider an example. One angle of a complementary pair of angles is x°, then, the other angle will be (90 – x)°.
In the figure,
∠AOD + ∠DOB
= 30° + 60°
= 90°
Hence, ∠AOD and ∠DOB form a complementary pair of angles.
When two lines intersect, four angles are formed. The two angles adjacent to each other are called supplementary angles. Such a pair of angles when added form the angle measure of a straight line as well, which is 180°, and hence these angles are called a linear pair.
In the figure below, lines 1 and 2 are intersecting lines. The four angles formed are ∠P, ∠Q, ∠R and ∠S.
Here, we can see that the supplementary pair of angles are:
- ∠P + ∠Q = 180° [Forming the straight line 1]
- ∠P + ∠S = 180° [Forming the straight line 2]
- ∠S + ∠R = 180° [Forming the straight line 1]
- ∠R + ∠Q = 180° [Forming the straight line 2]
Solved Examples
Example 1: Find the measure of the missing angle.
Solution:
The missing angle is ∠x.
∠AOC + ∠COB = 90 [Complementary angles formula]
32 + x = 90 [Substitute values]
x = 90 – 32 [Subtract 32 from both sides]
x = 68 [Subtract]
Hence, the measure of missing angle, x is 68°.
Example 2: Find the measure of missing angle.
Solution:
The missing angle is ∠x.
∠AOD + ∠DOB = 90 [Complementary angles formula]
∠AOD + (∠DOC + ∠COB) = 90
30 + (x + 25) = 90 [Substitute values]
(30 + 25) + x = 90 [Associative property of addition]
55 + x = 90 [Add]
x = 90 – 55 [Subtract 55 from both sides]
x = 35
Hence, the measure of missing angle x is 35°.
Example 3: Find the measure of missing angle.
Solution:
The missing angle is ∠y.
∠BOC + ∠AOC = 180 [Supplementary angle formula]
y + 70 = 180 [Substitute values]
y = 180 – 70 [Subtract 70 from both sides]
y = 110 [Subtract]
Hence, the measure of missing angle, y is 110°.
Example 4: Find the measure of missing angle.
Solution:
The missing angle is ∠y.
∠AOD + ∠DOB = 180 [Supplementary angles formula]
∠AOD + (∠DOC + ∠COB) = 180
y + (35 + 60) = 180 [Substitute values]
y + 95 = 180 [Add]
y = 180 – 95 [Subtract 95 from both sides]
y = 85
Hence, the measure of unknown angle y is 85°.
Example 5: Find the measure of missing angles.
Solution:
The missing angles are ∠x, ∠y and ∠z.
The two lines 1 and 2 are intersecting lines.
Hence,
110 + x = 180 [Supplementary angles]
x = 180 – 110 [Subtract 110 from both sides]
x = 70 [Subtract]
x + y = 180 [Supplementary angles]
70 + y = 180 [Substitute 70 for x]
y = 180 – 70 [Subtract 70 from both sides]
y = 110 [Subtract]
y + z = 180 [Supplementary angles]
110 + z = 180 [Substitute 110 for y]
z = 180 – 110 [Subtract 110 from both sides]
z = 70 [Subtract]
Hence, the measure of missing angles is x = 70°, y = 110° and z = 70°.
We know that an obtuse angle is greater than 90°, so the sum of two obtuse angles will always be greater than twice of 90°, that is greater than 180°. Hence, two obtuse angles cannot be supplementary angles.
We know that the measure of a right angle is 90°. So the sum of two right angles will be twice of 90°, that is, greater than 90°. Hence, two right angles cannot complement each other.
A pair of acute angles may or may not be complementary.
For example, acute angles 30° and 60° add up to 90°, so they are complementary angles.
However, acute angles 45° and 60° add up to 105°, so they are not complementary angles.
When two lines intersect, the angles that are vertically opposite to each other are called vertically opposite or vertical angles. Vertical angles are always equal in measure.
When the sum of two angles that are not adjacent, or they do not have a common vertex is 90°, then they are known as non-adjacent complementary angles.