## Complementary Angles

- Two acute angles form complementary angles.
- The sum of these two angle measures is
**90**^{o }(right angle). - Each angle complements the other.
- In the diagram shown below, $m\angle XOY+m\angle YOZ=90\xb0$, and hence they are complementary angles.

- Complementary angles do not have to be always adjacent angles. For example, $\angle AOB\mathrm{and}\angle CDE$ shown below are not adjacent angles, but still, they are complementary angles.

## Supplementary Angles

- Two angles are said to be supplementary if their sum measures $180\xb0$.
- In other words, two angles that form a line are called supplementary angles.
- Each angle
**supplements**the other angle. - In the diagram shown below, $m\angle AOB+m\angle BOC=180\xb0$, and hence they are a pair of supplementary angles.

- Supplementary angles do not have to be always adjacent angles. For example, $\angle XYZand\angle MOP$ shown below are not adjacent angles, but still, they are supplementary angles.

## Solved Examples

Question1: Find the measures of $\angle ABC$ and $\angle XYZ$ if they are complementary.

Solution: The sum of two complementary angles is $90\xb0$.

$\angle ABC+\angle XYZ=90\xb0$

$\frac{x}{4}+\frac{x}{5}=90\xb0$

$\frac{9x}{20}=90\xb0$

$x=\frac{90\xb0\times 20}{9}$

$x=200\xb0$

$\angle ABC=\frac{x}{4}$$=\frac{200\xb0}{4}$$=50\xb0$

$\angle XYZ=\frac{x}{5}$$=\frac{200\xb0}{5}$$=40\xb0$

Question 2: The difference between two supplementary angles is 20 degrees. Find the measure of both angles.

Solution: Let one of the angles be $x\xb0$. Then, the other angle will be $\left(180-x\right)\xb0$.

According to the question:

$\left(180-x\right)\xb0-x\xb0=20\xb0$

$180\xb0-2x\xb0=20\xb0$

$2x\xb0=180\xb0-20\xb0$

$2x\xb0=160\xb0$

$x\xb0=\frac{160\xb0}{2}$$=80\xb0$

So, one angle is $80\xb0$. Then, the other angle will be $100\xb0$.

## Cheat Sheet

- Complementary Angles:
- Two angles are said to be complementary if their sum is $90\xb0$.
- The complement of an angle $x\xb0$ is $\left(90-x\right)\xb0$.

- Supplementary Angles:

- Two angles are said to be supplementary if their sum is $180\xb0$.
- The supplement of an angle $y\xb0$ is $\left(180-y\right)\xb0$.

## Blunder Areas

- Two angles that add up to $90\xb0$ are complementary angles. Both angles are not necessarily adjacent.
- Two angles that add up to $180\xb0$ are supplementary angles. Both angles are not necessarily adjacent.

- Fiona Wong
- 10 Comments
- 57 Likes