Question

in the diagram, (\frac{vz}{yz} = \frac{wz}{xz})
diagram with points v, w, z, x, y forming two triangles intersecting at z
to prove that (\triangle vwz sim \triangle yxz) by the sas similarity theorem, which other sides or angles should be used?
(\bigcirc overline{wv}) and (overline{xy})
(\bigcirc overline{wv}) and (overline{zy})
(\bigcirc angle vzw cong angle yzx)
(\bigcirc angle vwz cong angle yxz)

Explanation:

To prove two triangles similar by the SAS (Side - Angle - Side) similarity theorem, we need two pairs of corresponding sides in proportion and the included angle between those sides congruent.

We are given that \(\frac{VZ}{YZ}=\frac{WZ}{XZ}\). Now, we need to check the included angle. The angle between \(VZ\) and \(WZ\) in \(\triangle VWZ\) is \(\angle VZW\), and the angle between \(YZ\) and \(XZ\) in \(\triangle YXZ\) is \(\angle YZX\). These two angles are vertical angles, and vertical angles are congruent. So, \(\angle VZW\cong\angle YZX\) is the included angle we need for the SAS similarity theorem.

Let's analyze the other options:

• Option 1: Considering \(\overline{WV}\) and \(\overline{XY}\) does not give us the included angle required for SAS similarity.
• Option 2: Considering \(\overline{WV}\) and \(\overline{ZY}\) is not relevant to the sides in the given proportion and does not help in applying SAS similarity.
• Option 4: \(\angle VWZ\cong\angle YXZ\) is not the included angle between the sides in proportion.

Answer:

\(\boldsymbol{\angle VZW \cong \angle YZX}\) (the third option: \(\boldsymbol{\angle VZW \cong \angle YZX}\))