Chapter 7.5, Problem 16WE

To determine

To Prove : For the given figure, $\overline{BC}\parallel \overline{AR}$

Explanation of Solution

Given information:

  $\frac{AB}{SR}=\frac{BC}{RA}=\frac{CA}{AS}$

  

According to SAS similarity theorem;

Now consider one triangle has vertices A, B, and C and another triangle has vertices S, R, and A.

  $\frac{AB}{SR}=\frac{BC}{RA}=\frac{CA}{AS}$

All the corresponding sides are in proportions,

Therefore, according to the SSS similarity theorem,

  $\text{Δ}ABC\sim \text{Δ}SRA$

Now if the lines BC and AR are parallel then the angles $\angle BCA\text{and}\angle RAS$ must be congruent with each other.

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

Thus,

  $\begin{array}{l}\angle ABC\cong \angle SRA\hfill \\ \angle CAB\cong \angle ASR\hfill \\ \angle BCA\cong \angle RAS\hfill \end{array}$

The angles $\angle BCA\text{and}\angle RAS$ are congruent with each other.

Therefore, the lines BC and AR are parallel.

Hence, $\overline{BC}\parallel \overline{AR}$ .