Chapter 4.6, Problem 8WE

a.

To determine

To list: key steps to prove $\overline{PR}$ bisects $\angle SQT$ .

Explanation of Solution

Given information: $\overline{PR}$ bisects $\angle SPT$ and $\angle SRT$

  

Step 1: Prove $\Delta PSR\cong \Delta PTR$

Step 2: Prove $\overline{SR}\cong \overline{TR}$

Step 3: Prove $\Delta SQR\cong \Delta TQR$

Step 4: Prove $\angle 3\cong \angle 4$

b.

To determine

To prove: $\overline{PR}$ bisects $\angle SQT$

Explanation of Solution

Given information: $\overline{PR}$ bisects $\angle SPT$ and $\angle SRT$

Proof:

  

Consider $\Delta PSR,\Delta PTR$

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

  $\angle 1=\angle 2$

  $\overline{PR}=\overline{PR}$

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

  $\angle 5=\angle 6$

So,

  $\Delta PSR\cong \Delta PTR\text{(ASA}\text{congruence}\text{criteria)}$

(according to ASA congruence criteria, if two angles and the included side of one triangle are congruent to the two angles and the included side of the other triangle then the triangles are said to be congruent)

So,

  $\overline{SR}=\overline{TR}(i)$ (Corresponding parts of congruent triangles are equal)

Now prove $\Delta SQR\cong \Delta TQR$

Consider $\Delta SQR,\Delta TQR$

  $\overline{SR}=\overline{TR}$ (proved in $(i)$ )

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

  $\angle 5=\angle 6$

  $\overline{QR}=\overline{QR}$

So,

  $\Delta SQR\cong \Delta TQR$ (SAS congruence criteria)

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

  $\angle 3=\angle 4$ (Corresponding parts of congruent triangles are equal)