Question

which step is missing? a. statement: \\(\frac{xw}{ut} = \frac{vw}{xv}\\) reason: corresponding sides of similar triangles are proportional. b. statement: \\(\frac{xw}{ut} = \frac{uv}{xv}\\) reason: corresponding sides of similar triangles are proportional. c. statement: \\(\frac{xw}{ut} = \frac{uv}{vw}\\) reason: corresponding sides of similar triangles are proportional. d. statement: \\(\frac{xw}{ut} = \frac{xv}{uv}\\) reason: corresponding sides of similar triangles are proportional. statements: \\(\overline{tu} \parallel \overline{xw}\\) (given), \\(\angle utv \cong \angle xwv\\) (alternate angles theorem), \\(\angle tvu \cong \angle wvx\\) (vertical angles theorem), \\(\triangle tvu \sim \triangle wvx\\) (aa),?, \\(\frac{xw}{14} = \frac{4}{7}\\) (substitution property of equality), \\(7(xw) = 56\\) (cross product), \\(xw = 8\\) (division property of equality)

Explanation:

Step1: Analyze the Similar Triangles

Since \(\triangle TVU\sim\triangle WVX\) (by AA similarity), the corresponding sides of similar triangles are proportional. The correct proportion for the sides should be \(\frac{TU}{XW}=\frac{UV}{VW}=\frac{TV}{VX}\).

Step2: Check Each Option

• Option A: \(\frac{TU}{XW}=\frac{VY}{VW}\) is incorrect. The correct proportion for the sides of similar triangles \(\triangle TVU\) and \(\triangle WVX\) should have corresponding sides. \(VY\) is not a side of either triangle in the similarity relation.
• Option B: \(\frac{TU}{XW}=\frac{UV}{XV}\) is incorrect. \(XV\) is not a side that corresponds to \(UV\) in the similar triangles \(\triangle TVU\) and \(\triangle WVX\).
• Option C: \(\frac{TU}{XW}=\frac{UV}{VW}\) is correct. In similar triangles \(\triangle TVU\sim\triangle WVX\), \(TU\) corresponds to \(XW\) and \(UV\) corresponds to \(VW\).
• Option D: \(\frac{TU}{XW}=\frac{XV}{UV}\) is incorrect. The ratio of sides should follow the order of corresponding sides in the similar triangles.

Answer:

C. Statement: \(\frac{TU}{XW}=\frac{UV}{VW}\) Reason: Corresponding sides of similar triangles are proportional.