Question

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

$\mathbf{m}\angle \mathbf{C}\mathbf{=}\mathbf{3}\mathbf{y}\mathbf{+}\mathbf{5}$,$\mathbf{m}\angle \mathbf{D}\mathbf{=}\mathbf{2}\mathbf{y}$

Short answer

The value of y is 17.

The measures of the angles C and D are 56° and 34°.

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,

$$\begin{gathered} m\angle C+m\angle D=90° \\ 3y+5+2y=90 \\ 5y+5=90 \\ 5y=90-5 \\ 5y=85 \\ y=\frac{85}{5} \\ y=17 \end{gathered}$$

Therefore, the value of y is 17.

Step 4. Find the measure of the angle C.

Substitute 17 for y into $m\angle C=3y+5$ to find the measure of the angle C.

Therefore,

$$\begin{gathered} m\angle C=3y+5 \\ =3\times 17+5 \\ =51+5 \\ =56 \end{gathered}$$

Therefore, the measure of angleC is 56°.

Step 5. Find the measure of the angle D.

Substitute 17 fory into$m\angle D=2y$ to find the measure of the angle D.

Therefore,

$$\begin{gathered} m\angle D=2y \\ \begin{array}{c}=2\times 17\\ =34\end{array} \end{gathered}$$

Therefore, the measure of the angle D is 34°.