Question

Describe your plan for proving the following.

1. Given:$\overline{\mathbf{P}\mathbf{R}}$ bisects$\mathbf{\angle }\mathit{Q}\mathit{P}\mathit{S}\mathbf{;}\overline{\mathbf{P}\mathbf{Q}}\mathbf{\cong }\overline{\mathbf{P}\mathbf{S}}$ Prove:$\mathbf{\angle }\mathit{Q}\mathbf{\cong }\mathbf{\angle }\mathit{S}$

Short answer

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

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

$\Delta PQR\cong \Delta SPR$(SAS congruence criteria)

$\angle Q\cong \angle S$ (corresponding parts of congruent triangles)

Step-by-step solution

- Show

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

- Show ∠QPR≅∠SPR

As bisector of an angle divides it into congruent angles

- Show ΔPQR≅ΔPSR

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

So, from step 1, step 2 and using given

$\Delta PQR\cong \Delta SPR$

- Show

As corresponding parts of congruent triangles are equal

Hence,$\angle Q\cong \angle S$