Question

which of the following best completes the proof showing that \\(\delta wxz \sim \delta xyz\\)?

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.

Explanation:

Identify the given side lengths

We are given the lengths of the segments in the figure:

• \(WZ = 10\)
• \(XZ = 5\)
• \(ZY = 2.5\)

Determine corresponding sides for similarity

We want to show that \(\Delta WXZ \sim \Delta XYZ\).
The corresponding sides adjacent to the congruent right angles \(\angle WZX\) and \(\angle XZY\) are:

• For \(\Delta WXZ\): sides \(WZ\) and \(XZ\)
• For \(\Delta XYZ\): sides \(XZ\) and \(YZ\)

Set up the proportion

We compare the ratios of the corresponding sides:

$$ \frac{WZ}{XZ} = \frac{10}{5} = 2 $$

$$ \frac{XZ}{YZ} = \frac{5}{2.5} = 2 $$

Thus, the proportion that shows the corresponding sides are proportional is:

$$ \frac{WZ}{XZ} = \frac{XZ}{YZ} $$

Substituting the values:

$$ \frac{10}{5} = \frac{5}{2.5} $$

This completes the proof using the SAS Similarity Postulate.

Answer:

Since \(\overline{XZ} \perp \overline{WY}\), angles WZX and XZY are both right angles and congruent. The proportion <blank>\(\frac{WZ}{XZ} = \frac{XZ}{YZ}\)</blank> shows the corresponding sides are proportional, so the triangles are similar by the SAS Similarity Postulate.