Question

$\overline{wy}$ is an altitude in triangle $wxz$. if $\triangle ywz \sim \triangle yxw$, what is true about $\angle xwz$? $\bigcirc$ $\angle xwz$ is an obtuse angle. $\bigcirc$ $\angle xwz$ is a right angle. $\bigcirc$ $\angle xwz$ is congruent to $\angle wxy$. $\bigcirc$ $\angle xwz$ is congruent to $\angle xzw$.

Explanation:

Step1: Use similarity angle correspondence

Since $\triangle YWZ \sim \triangle YXW$, corresponding angles are congruent: $\angle WZY = \angle XWY$ and $\angle ZWY = \angle WXY$.

Step2: Use altitude right angle

$\overline{WY}$ is an altitude, so $\angle WYX = \angle WYZ = 90^\circ$.

Step3: Sum angles in $\triangle WXZ$

In $\triangle WXZ$, $\angle XWZ = \angle XWY + \angle ZWY$. Substitute congruent angles: $\angle XWZ = \angle WZY + \angle WXY$.
The sum of interior angles in $\triangle WXZ$ is $180^\circ$:
$$\angle XWZ + \angle WXY + \angle WZY = 180^\circ$$
Substitute $\angle WZY + \angle WXY = \angle XWZ$:
$$\angle XWZ + \angle XWZ = 180^\circ$$
$$2\angle XWZ = 180^\circ$$
$$\angle XWZ = 90^\circ$$

Answer:

$\angle XWZ$ is a right angle.