11. if $ riangle zmk cong riangle apy$, $mangle m = 112^{circ}$, $mangle y = 41^{circ}$, $mangle k=(13x - 37)^{circ}$, and $mangle a=(2y + 7)^{circ}$, find the values of $x$ and $y$.

Question

11. if $	riangle zmk cong 	riangle apy$, $mangle m = 112^{circ}$, $mangle y = 41^{circ}$, $mangle k=(13x - 37)^{circ}$, and $mangle a=(2y + 7)^{circ}$, find the values of $x$ and $y$.

Accepted Answer

# Explanation:
## Step1: Use the property of congruent triangles
Since $\triangle ZMK\cong\triangle APY$, then $\angle K=\angle Y$ and $\angle A = \angle Z$.
Given $m\angle Y = 41^{\circ}$, so $m\angle K=(13x - 37)^{\circ}=41^{\circ}$
$$
\begin{align*}
13x-37&=41\
13x&=41 + 37\
13x&=78\
x&=\frac{78}{13}\
x& = 6
\end{align*}
$$
## Step2: Find the measure of $\angle Z$
In $\triangle ZMK$, using the angle - sum property of a triangle ($m\angle Z+m\angle M+m\angle K=180^{\circ}$).
We know $m\angle M = 112^{\circ}$ and $m\angle K = 41^{\circ}$
$$
\begin{align*}
m\angle Z&=180-(112 + 41)\
m\angle Z&=180 - 153\
m\angle Z&=27^{\circ}
\end{align*}
$$
Since $m\angle A=(2y + 7)^{\circ}$ and $m\angle A=m\angle Z$ (because $\triangle ZMK\cong\triangle APY$)
$$
\begin{align*}
2y+7&=27\
2y&=27 - 7\
2y&=20\
y&=\frac{20}{2}\
y&=10
\end{align*}
$$

# Answer:
$x = 6$ and $y = 10$

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QUESTION IMAGE

Question

Explanation:

Step1: Use the property of congruent triangles

Step2: Find the measure of \(\angle Z\)

Answer: