Chapter 4.3, Problem 2CE

To determine

Toprove: $\overline{RQ}\cong \overline{RS}$

Explanation of Solution

Given information: $\overline{PR}$ bisects $\angle QPS$ and $\angle QRS$

Proof:

  

Consider $\Delta PRQ,\Delta PRS$

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

  $\angle QPR\cong \angle SPR$ (angle bisector bisects the angle into two congruent angles)

Also,

  $\overline{PR}\cong \overline{PR}$ (common sides of $\Delta PRQ,\Delta PRS$ )

As $\overline{PR}$ bisects $\angle QRS$ ,

  $\angle QRP\cong \angle SRP$ (angle bisector bisects the angle into two congruent angles)

Therefore,

  $\Delta PRQ\cong \Delta PRS$ (by ASA congruence criteria)

Now, if two triangles are congruent then the corresponding angles and corresponding sides are also congruent.

As $\Delta PRQ\cong \Delta PRS$ ,

  $\overline{RQ}\cong \overline{RS}$ ( $\overline{RQ},\overline{RS}$ are corresponding sides of $\Delta PRQ,\Delta PRS$ )