Question

use the given information to find the indicated values. 9. given $\triangle abccong\triangle def$, find the values of $x$ and $y$.

Explanation:

Step1: Use the property of congruent triangles for angles

Since $\triangle ABC\cong\triangle DEF$, corresponding angles are equal. $\angle A=\angle D$. So, $87^{\circ}=(5x + 2)^{\circ}$.
\[

$$\begin{align*} 5x+2&=87\\ 5x&=87 - 2\\ 5x&=85\\ x&=\frac{85}{5}\\ x& = 17 \end{align*}$$

\]

Step2: Use the angle - sum property of a triangle

In $\triangle ABC$, the sum of interior angles of a triangle is $180^{\circ}$. So, $\angle A+\angle B+\angle C=180^{\circ}$. In $\triangle DEF$, $\angle D+\angle E+\angle F = 180^{\circ}$. Since $\triangle ABC\cong\triangle DEF$, we can also use the fact that $\angle C=\angle F$. Let's find $\angle C$ in $\triangle ABC$. First, find $\angle B$ in $\triangle ABC$. In $\triangle DEF$, $\angle E = 42^{\circ}$, so $\angle B=42^{\circ}$ (corresponding angles of congruent triangles). Then in $\triangle ABC$, $\angle C=180^{\circ}-\angle A-\angle B=180 - 87-42=51^{\circ}$. So, $3y = 51$.
\[

$$\begin{align*} 3y&=51\\ y&=\frac{51}{3}\\ y&=17 \end{align*}$$

\]

Answer:

$x = 17$, $y = 17$