Question

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

Explanation:

Step1: Recall Similar Triangles Property

In similar triangles, corresponding angles are congruent. Given \( \triangle YWZ \sim \triangle YXW \), so \( \angle YWZ \cong \angle YXW \) and \( \angle YZW \cong \angle YWX \), and \( \angle Y \) is common (right angle as \( WY \) is altitude).

Step2: Analyze Triangle \( WXZ \)

Since \( WY \) is an altitude, \( \angle WYX = \angle WYZ = 90^\circ \). From similarity, let's relate angles in \( \triangle WXZ \). We know that in a triangle, if two angles relate via similarity, we can check the right angle. Also, from the similarity \( \triangle YWZ \sim \triangle YXW \), we can infer about \( \angle XWZ \). A right angle in similar triangles and the altitude implies that \( \angle XWZ \) must be a right angle because the triangles formed by the altitude in a right - angled triangle (or when similarity implies right angle) will have \( \angle XWZ \) as right angle. Also, let's check other options:

• Option 1: \( \angle XWZ \cong \angle XZW \) is not necessarily true from similarity.
• Option 2: \( \angle XWZ \) is not obtuse as the altitude and similarity suggest right angle.
• Option 4: \( \angle XWZ \cong \angle WXY \) is not correct from similarity correspondence.

Answer:

\( \angle XWZ \) is a right angle. (The option: \( \angle XWZ \) is a right angle.)