Question

given: $overline{ps}paralleloverline{qr}$, $angle qpscongangle srq$. prove: $overline{pq}congoverline{rs}$. statements 1. $overline{ps}paralleloverline{qr}$, $angle qpscongangle srq$ 2. $overline{qs}congoverline{qs}$ 3. $angle psqcongangle sqr$ 4. $\triangle psqcong\triangle rqs$ 5. $overline{pq}congoverline{rs}$. reasons 1. given 2. 3. 4. 5.

Explanation:

Step1: Reflexive property

Any segment is congruent to itself, so $\overline{QS}\cong\overline{QS}$.

Step2: Alternate - interior angles

Since $\overline{PS}\parallel\overline{QR}$, the alternate - interior angles $\angle PSQ$ and $\angle SQR$ are congruent.

Step3: Angle - Side - Angle (ASA) congruence

We have $\angle QPS\cong\angle SRQ$ (given), $\overline{QS}\cong\overline{QS}$ (reflexive property), and $\angle PSQ\cong\angle SQR$ (alternate - interior angles), so $\triangle PSQ\cong\triangle RQS$ by ASA.

Step4: Corresponding parts of congruent triangles

If $\triangle PSQ\cong\triangle RQS$, then the corresponding sides $\overline{PQ}$ and $\overline{RS}$ are congruent.

Answer:

1. Reflexive property
2. Alternate - interior angles
3. ASA (Angle - Side - Angle)
4. Corresponding parts of congruent triangles (CPCTC)