Question

which of the following best completes the proof showing that δwxz ~ δxyz?
diagram: triangle wxy with z on wy, xz perpendicular to wy, wz=16, zy=9, xz=12, right angle at z
since (overline{xz} perp overline{wy}), angles wzx and xzy are both right angles and congruent. the proportion ______ shows the corresponding sides are proportional, so the triangles are similar by the sas similarity postulate.
options:
(\frac{16}{12} = \frac{9}{12})
(\frac{16}{12} = \frac{12}{9})
(\frac{12}{12} = \frac{16}{9})
(\frac{12}{16} = \frac{16}{9})

Explanation:

Step1: Identify corresponding sides

In similar triangles \( \triangle WXZ \) and \( \triangle XYZ \) (wait, actually \( \triangle WXZ \) and \( \triangle XYZ \)? Wait, the problem is \( \triangle WXZ \sim \triangle XYZ \)? Wait, no, the problem says \( \triangle WXZ \sim \triangle XYZ \)? Wait, the diagram: \( XZ \perp WY \), so \( \angle WZX = \angle XZY = 90^\circ \). For SAS similarity, we need two sides in proportion and included angle equal. The included angle is the right angle. So the sides adjacent to the right angle: in \( \triangle WXZ \), the sides are \( WZ = 16 \), \( XZ = 12 \); in \( \triangle XYZ \), the sides are \( XZ = 12 \), \( ZY = 9 \). So the proportion should be \( \frac{WZ}{XZ} = \frac{XZ}{ZY} \), which is \( \frac{16}{12} = \frac{12}{9} \).

Step2: Check the options

Looking at the options, the second option is \( \frac{16}{12} = \frac{12}{9} \), which matches the proportion we derived for SAS similarity (since the included angle \( \angle WZX = \angle XZY = 90^\circ \) is congruent).

Answer:

\(\boldsymbol{\frac{16}{12} = \frac{12}{9}}\) (the second option, e.g., if options are labeled, say B. \(\frac{16}{12} = \frac{12}{9}\))