Question

in the diagram, \\( \frac { v z } { y z } = \frac { w z } { x z } \\) to prove that \\( \triangle v w z - \triangle y x z \\) by the sas similarity theorem, which other sides or angles should be used? \\( \overline { w v } \\) and \\( \overline { x y } \\) \\( \overline { w v } \\) and \\( \overline { z y } \\) \\( \angle v z w \cong \angle y z x \\) \\( \angle v w z \cong \angle y x z \\)

Explanation:

Step1: Recall the SAS similarity theorem

The SAS (Side - Angle - Side) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the two triangles are similar.

Step2: Analyze the given proportion

We are given that \( \frac{VZ}{YZ}=\frac{WZ}{XZ} \). The included angles for these sides in \( \triangle VWZ \) and \( \triangle YXZ \) are \( \angle VZW \) and \( \angle YZX \) respectively.

Step3: Check the options

• Option A: \( \overline{WV} \) and \( \overline{XY} \) are not related to the given proportion \( \frac{VZ}{YZ}=\frac{WZ}{XZ} \) for the SAS similarity.
• Option B: \( \overline{WV} \) and \( \overline{ZY} \) are not relevant to the SAS similarity based on the given proportion.
• Option C: If \( \angle VZW\cong\angle YZX \), along with \( \frac{VZ}{YZ}=\frac{WZ}{XZ} \), by the SAS similarity theorem \( \triangle VWZ\sim\triangle YXZ \).
• Option D: \( \angle VWZ \) and \( \angle YXZ \) are not the included angles for the sides \( VZ,YZ,WZ,XZ \) given in the proportion \( \frac{VZ}{YZ}=\frac{WZ}{XZ} \).

Answer:

C. \( \angle VZW\cong\angle YZX \)