If the opposite sides are parallel .. and then the angle P= Q
AND
ANGLE R = S.

Proof:

To prove that angle P = angle Q and angle R = angle S, we can use the properties of parallel lines and transversals.

Given: PQ || RS and PS = QR

Proof of angle P = angle Q:

1. Since PQ || RS, we can say that angle PQS and angle QRS are corresponding angles.
2. Corresponding angles are congruent when the lines are parallel.
3. Therefore, angle PQS = angle QRS.
4. Since PS = QR, we can also say that angle QPS = angle QRP by vertically opposite angles theorem.
5. Now, we have angle PQS = angle QRS and angle QPS = angle QRP.
6. By adding these two equations, we get angle PQS + angle QPS = angle QRS + angle QRP.
7. Simplifying the equation, we get angle PQS + angle QPS = angle QRP + angle QRS.
8. Rearranging the terms, we get angle P + angle Q = angle R + angle S.
9. Since angle PQS = angle QRS, we can substitute angle P + angle Q with angle R + angle S.
10. Therefore, angle P + angle Q = angle R + angle S becomes angle R + angle S = angle R + angle S.
11. This implies that angle P = angle Q.

Proof of angle R = angle S:

1. Since PQ || RS, we can say that angle PSQ and angle QSR are corresponding angles.
2. Corresponding angles are congruent when the lines are parallel.
3. Therefore, angle PSQ = angle QSR.
4. Since PS = QR, we can also say that angle SPQ = angle SRQ by vertically opposite angles theorem.
5. Now, we have angle PSQ = angle QSR and angle SPQ = angle SRQ.
6. By adding these two equations, we get angle PSQ + angle SPQ = angle QSR + angle SRQ.
7. Simplifying the equation, we get angle PSQ + angle SPQ = angle SRQ + angle QSR.
8. Rearranging the terms, we get angle R + angle S = angle R + angle S.
9. This implies that angle R = angle S.

Therefore, we have proved that angle P = angle Q and angle R = angle S using the given conditions and properties of parallel lines and transversals.