Question

pr bisects ∠qps and ∠qrs. complete the proof that △pqr ≅ △psr.
1 pr bisects ∠qps
2 pr bisects ∠qrs
3 ∠qpr ≅ ∠rps
4 ∠prq ≅ ∠prs
5 pr ≅ pr
6 △pqr ≅ △psr

Explanation:

Step1: Given information

1. \( \overleftrightarrow{PR}\) bisects \( \angle QPS\) - Given
2. \( \overleftrightarrow{PR}\) bisects \( \angle QRS\) - Given

Step2: Angle - bisector property

1. \( \angle QPR\cong\angle RPS\) - Definition of angle bisector (If a ray bisects an angle, it divides the angle into two congruent angles)
2. \( \angle PRQ\cong\angle PRS\) - Definition of angle bisector

Step3: Reflexive property

1. \( \overline{PR}\cong\overline{PR}\) - Reflexive property of congruence (Any segment is congruent to itself)

Step4: ASA (Angle - Side - Angle) congruence criterion

1. \( \triangle PQR\cong\triangle PSR\) - ASA congruence rule (\( \angle QPR\cong\angle RPS\), \( \overline{PR}\cong\overline{PR}\), \( \angle PRQ\cong\angle PRS\))

Answer:

1. Given
2. Given
3. Definition of angle bisector
4. Definition of angle bisector
5. Reflexive property of congruence
6. ASA (Angle - Side - Angle) congruence criterion