Chapter 4.3, Problem 1CE

To determine

To prove: $\angle Q\cong \angle S$

Explanation of Solution

Given information: $\overleftrightarrow{PR}$ bisects $\angle QPS$ , $\overline{PQ}\cong \overline{PS}$

Proof:

First prove $\Delta PQR\cong \Delta PSR$

Consider $\Delta PQR,\Delta PSR$

  

According to reflexive property of congruence, a line segment is congruent to itself.

  $\overline{PR}\cong \overline{PR}$ (reflexive property of congruence)

As $\overleftrightarrow{PR}$ bisects $\angle QPS$ ,

  $\angle QPR\cong \angle SPR$ (bisector of an angle divides it into congruent angles)

  $\overline{PQ}\cong \overline{PS}$ (Given)

So,

  $\Delta PQR\cong \Delta PSR$ (By SAS congruence criteria)

(According to SAS congruence criteria, two triangles are said to be congruent if two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle)

Now, as corresponding parts of congruent triangles are equal, $\angle Q\cong \angle S$ .