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.

choices:

• \\(\frac{5}{10} = \frac{5}{2.5}\\)
• \\(\frac{5}{10} = \frac{5}{5}\\)
• \\(\frac{10}{5} = \frac{5}{2.5}\\)
• \\(\frac{10}{5} = \frac{2.5}{5}\\)

Explanation:

Identify corresponding sides

To prove \(\Delta WXZ \sim \Delta XYZ\) using the SAS Similarity Postulate, we need to show that the two pairs of sides adjacent to the congruent right angles (\(\angle WZX \cong \angle XZY\)) are proportional.
The sides forming \(\angle WZX\) in \(\Delta WXZ\) are \(WZ\) and \(XZ\).
The sides forming \(\angle XZY\) in \(\Delta XYZ\) are \(XZ\) and \(YZ\).

Set up the proportion

We match the corresponding sides of \(\Delta WXZ\) and \(\Delta XYZ\):

• The longer leg of \(\Delta WXZ\) is \(WZ = 10\), and the longer leg of \(\Delta XYZ\) is \(XZ = 5\).
• The shorter leg of \(\Delta WXZ\) is \(XZ = 5\), and the shorter leg of \(\Delta XYZ\) is \(YZ = 2.5\).

Thus, the ratio of corresponding sides is:

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

Substitute the given values

Substitute the known lengths \(WZ = 10\), \(XZ = 5\), and \(YZ = 2.5\) into the proportion:

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

Both ratios simplify to \(2\), confirming that the corresponding sides are proportional. This matches the third option.

Answer:

• (A) \(\frac{5}{10} = \frac{5}{2.5}\)
• (B) \(\frac{5}{10} = \frac{5}{5}\)
• (C) \(\frac{10}{5} = \frac{5}{2.5}\) (Correct answer)
• (D) \(\frac{10}{5} = \frac{2.5}{5}\)