Question

given that $\frac{vw}{vx}=\frac{vz}{vy}$, prove $overline{wz}paralleloverline{xy}$.

Explanation:

Step1: Recall similarity - side - ratio criterion

If in two triangles, the ratios of the lengths of corresponding sides are equal, the triangles are similar. In \(\triangle VWZ\) and \(\triangle VXY\), we are given that \(\frac{VW}{VX}=\frac{VZ}{VY}\), and \(\angle V\) is common to both triangles. So, \(\triangle VWZ\sim\triangle VXY\) by the Side - Angle - Side (SAS) similarity criterion (since \(\angle V=\angle V\) and the side - ratios are equal).

Step2: Use corresponding - angle property of similar triangles

For similar triangles \(\triangle VWZ\) and \(\triangle VXY\), corresponding angles are equal. Specifically, \(\angle VWZ=\angle VXY\) (corresponding angles of similar triangles).

Step3: Apply parallel - line criterion

If two lines are cut by a transversal and the corresponding angles are equal, then the two lines are parallel. Here, \(\overline{WZ}\) and \(\overline{XY}\) are cut by the transversal \(\overline{VX}\), and since \(\angle VWZ=\angle VXY\), we can conclude that \(\overline{WZ}\parallel\overline{XY}\).

Answer:

We have proven that \(\overline{WZ}\parallel\overline{XY}\) using the similarity of triangles and the corresponding - angle property for parallel lines.