Chapter 7.5, Problem 14WE

To determine

To choose: The given option which is correct.

Answer to Problem 14WE

The $\angle \text{l}\cong \angle Y$ must be true.

Explanation of Solution

Given information:

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

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

The triangles $VWX$ and $VYX$ 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,

  $\text{Δ}VWZ\sim \text{Δ}VXY$

(1)

According to the SAS similarity theorem,

  $\text{Δ}VWZ\sim \text{Δ}VYX$

The arrangement of the vertices does matter.

Therefore, if both the triangles are equilateral triangle only then triangle $VWZ$ will be similar to the triangle $VXY$ because only then the arrangement of vertices will not affect the proportion of the corresponding sides.

Hence, $\text{Δ}VWZ\sim \text{Δ}VXY$ may not be true.

(2)

According to the SAS similarity theorem,

  $\text{Δ}VWZ\sim \text{Δ}VYX$

The arrangement of the vertices does matter therefore, if both the triangles are equilateral triangle only then $\angle W$ and $\angle Z$ of the triangle $VWZ$ will be congruent to $\angle X$ and $\angle Y$ respectively, the angles of triangle $VYX,$ or else all the corresponding angles will not be congruent.

Therefore, according to the given conditions,

  $\angle W\cong \angle Y$

  $\angle Z\cong \angle X$

Therefore, the lines $WZ$ and $VY$ are may not be parallel in nature.

Hence, $\overline{WS}\parallel \overline{XY}$ may not be true.

(3)

According to the SAS similarity theorem,

  $\text{Δ}VWZ\sim \text{Δ}VYX$

Therefore, the corresponding angles should also be congruent.

  $\angle W\cong \angle Y\text{ or }\angle 1\cong \angle Y$

  $\angle Z\cong \angle Z\text{ or }\angle 2\cong \angle Z$

Hence, $\angle \text{l}\cong \angle Y$ must be true.