Question

If $\angle \mathbf{C}$and $\angle \mathbf{D}$are complementary, find the value of y, $\mathbf{m}\angle \mathbf{C}$and $\mathbf{m}\angle \mathbf{D}$.

$\mathbf{m}\angle \mathbf{C}\mathbf{=}\mathbf{y}\mathbf{-}\mathbf{8}$,$\mathbf{m}\angle \mathbf{D}\mathbf{=}\mathbf{3}\mathbf{y}\mathbf{+}\mathbf{2}$

Short answer

The value of y is 24.

The measures of the angles C and D are 16° and 74°.

Step-by-step solution

Step 1. Definition of complementary angles.

The complementary angles are the angles whose sum is 90°.

Step 2. Write the relation between the angles C and D.

As the angles C andD are complementary, therefore, the sum of the angles C andD is 90°.

Therefore, the relation between the angles C and D is:$m\angle C+m\angle D=90°$

Step 3. Find the value of y.

Find the value of y by using the relation $m\angle C+m\angle D=90°$.

Therefore,

$$\begin{gathered} m\angle C+m\angle D=90° \\ y-8+3y+2=90 \\ 4y-6=90 \\ 4y=90+6 \\ 4y=96 \\ y=\frac{96}{4} \\ y=24 \end{gathered}$$

Therefore, the value of y is 24.

Step 4. Find the measure of the angle C.

Substitute 24 for y into $m\angle C=y-8$ to find the measure of the angle C.

Therefore,

$$\begin{gathered} m\angle C=y-8 \\ \begin{array}{c}=24-8\\ =16\end{array} \end{gathered}$$

Therefore, the measure of angle C is 16°.

Step 5. Find the measure of the angle D.

Substitute 24 foryinto$m\angle D=3y+2$ to find the measure of the angle D.

Therefore,

$$\begin{gathered} m\angle D=3y+2 \\ =3\times 24+2 \\ =72+2 \\ =74 \end{gathered}$$

Therefore, the measure of the angle D is 74°.