Question

in the diagram, \\( \frac { v z } { y z } = \frac { w z } { x z } \\).
to prove that \\( \triangle v w z \sim \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 SAS similarity theorem

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 \(\frac{VZ}{YZ}=\frac{WZ}{XZ}\). The included angle for sides \(VZ\) and \(WZ\) in \(\triangle VWZ\) is \(\angle VZW\), and the included angle for sides \(YZ\) and \(XZ\) in \(\triangle YXZ\) is \(\angle YZX\). Since \(\angle VZW\) and \(\angle YZX\) are vertical angles and vertical angles are congruent (\(\angle VZW\cong\angle YZX\)).

Answer:

\(\angle VZW\cong\angle YZX\)