Chapter 7, Problem 2PE

To determine

To prove: $\overrightarrow{SQ}$ bisects $\angle RST$

Explanation of Solution

Given information: $\overrightarrow{QS}$ bisects $\angle RQT$ , $\angle R\cong \angle T$

Proof:

  

Consider $\Delta RQS,\Delta TQS$

  $\angle R\cong \angle T\text{(Given)}$

  $\angle RQS\cong \angle TQS$ (As $\overrightarrow{QS}$ bisects $\angle RQT$ )

  $\overline{QS}\cong \overline{SQ}$ (according to reflexive property of congruence, if $l$ is a line segment then $l\cong l$ )

So,

  $\Delta RQS\cong \Delta TQS$

(according to AAS congruence criteria, if two angles and the side not included between the angles of two triangles are congruent to each other then the two triangles are said to be congruent)

Therefore,

  $\angle RSQ\cong \angle TSQ$ (corresponding angles of congruent triangles are congruent)