Chapter 7.5, Problem 13WE

To determine

To proof: The sides of triangle is $\overline{WS}\parallel \overline{XY}$ .

Answer to Problem 13WE

The lines $WZ$ and $VY$ are parallel to each other. So, $\overline{WS}\parallel \overline{XY}$ is proved.

Explanation of Solution

Given information:

The system given is: $\frac{VW}{VX}=\frac{VZ}{VY}$

Explaination:

Now consider one triangle has vertices $V,W,$ and $Z$ another triangle has vertices $V,X,$ and Y.

Both the triangles have a common angle, $V$

And,

  $\frac{VW}{VX}=\frac{VZ}{VY}$

The corresponding sides are in proportions,

Therefore, according to the SAS similarity theorem;

Therefore,

  $\text{Δ}VWZ\sim \text{Δ}VXY$ .

As the triangles are similar, their remaining corresponding angles also must be congruent with each other.

Thus,

  $\angle W\cong \angle X$

  $\angle Z\cong \angle Y$

According to the definition, when a line crosses two parallel lines, the corresponding angles line makes with the parallel lines must be congruent.

And, in this case, a straight line $VX$ crosses two lines $WZ$ and $XY$ makes two angles, $\angle W$ and $\angle X$ .

And these two angles are congruent with each other; $\angle W\cong \angle X$ because they are corresponding angles of two similar triangles then lines must be parallel in nature.

Therefore, the lines $WZ$ and $VY$ are parallel.

Hence, $\overline{WS}\parallel \overline{XY}$ is proved.