Chapter 5.1, Problem 30WE

To determine

To prove: that $\angle P\cong \angle R$ with the help of given information.

Answer to Problem 30WE

  $\angle P\cong \angle R$ by side angle side property the triangles are congruent.

Explanation of Solution

Given information:

The quadrilateral JQKS is a parallelogram; $PJ\cong RK$

System diagram:

  

Calculation:

To prove $\angle P\cong \angle R$

It is known that,

  $PJ\cong RK\cdots \cdots (1)$

The opposite sides of the parallelogram are parallel

  $SK\parallel JK$

And opposite angles of a parallelogram are equal

  $\angle KSJ\cong \angle KQJ\dots \dots (2)$

And

  $\begin{array}{l}\angle 1\cong \angle KSJ\\ \angle 2=\angle KQJ\mathrm{....}(3)\end{array}$

From equation (2) and (3)

  $\angle \text{l}\cong \angle 2\dots \dots (4)$

Consider In $△SPJ$ and $△KRQ$

Since opposite sides of a parallelogram are equal

Thus,

  $PJ\cong RK$

  $SK\cong JK$

And

  $\angle \text{l}\cong \angle 2$

Thus, by side angle side property the triangles are congruent.

  $\text{Δ}SPJ\cong △KRQ$

Thus, by congruent parts of corresponding triangles $\angle P\cong \angle R$

Hence, $\angle P\cong \angle R$ by side angle side property the triangles are congruent.