Question

wy is an altitude in triangle wxz. if δywz ~ δyxw, what is true about ∠xwz? ∠xwz is an obtuse angle. ∠xwz is a right angle. ∠xwz is congruent to ∠wxy. ∠xwz is congruent to ∠xzw

Explanation:

Step1: Recall Similar Triangles Property

In similar triangles, corresponding angles are congruent. Given \( \triangle YWZ \sim \triangle YXW \), so \( \angle WYZ \) corresponds to \( \angle XYW \), \( \angle YWZ \) corresponds to \( \angle YXW \), and \( \angle YZW \) corresponds to \( \angle YWX \). Also, \( \overline{WY} \) is an altitude, so \( \angle WYX = 90^\circ \) (since altitude is perpendicular to the base).

Step2: Analyze Angles in Triangle \( WXZ \)

From the similarity \( \triangle YWZ \sim \triangle YXW \), we know that \( \angle YWX=\angle YZW \) and \( \angle YXW = \angle YWZ \). Now, in triangle \( WXZ \), we know that \( \angle WYX = 90^\circ \) (because \( WY \) is an altitude). Let's consider the angles of \( \triangle WXZ \). The sum of angles in a triangle is \( 180^\circ \). Also, since \( \triangle YWZ \sim \triangle YXW \), we can deduce that \( \angle XWZ \) is a right angle. Let's see: \( \angle XWZ=\angle XWY + \angle YWZ \). From similarity, \( \angle YWZ=\angle YXW \), and \( \angle XWY + \angle YXW=90^\circ \) (because \( \angle WYX = 90^\circ \) in right triangle \( WYX \)). Wait, actually, more directly: since \( \triangle YWZ \sim \triangle YXW \), \( \angle WYZ=\angle XYW \). But \( \angle WYZ + \angle XYW = 180^\circ \) (linear pair), so each must be \( 90^\circ \). Wait, no, \( \overline{WY} \) is an altitude, so \( \angle WYX = 90^\circ \) and \( \angle WYZ = 90^\circ \). Now, from similarity \( \triangle YWZ \sim \triangle YXW \), \( \angle YWZ=\angle YXW \) and \( \angle YZW=\angle YWX \). Now, in \( \triangle WXZ \), \( \angle XWZ=\angle XWY+\angle YWZ \). But \( \angle XWY + \angle YXW = 90^\circ \) (since \( \angle WYX = 90^\circ \)), and \( \angle YWZ=\angle YXW \), so \( \angle XWZ=\angle XWY+\angle YXW = 90^\circ \). So \( \angle XWZ \) is a right angle.

Answer:

\( \angle XWZ \) is a right angle.