Question

The figure shows quadrilateral P Q R S. Diagonals segment P R and segment Q S intersect at point T.



A picture of quadrilateral P Q R S.

Question
Choose ,begin emphasis,all,end emphasis, the statements that contain enough information to prove that P Q R S is a parallelogram.

Answer options with 4 options
A.
triangle P Q R is congruent to triangle R S P

B.
segment P S is congruent to segment Q R and segment P Q is parallel to segment S R

C.
angle S P Q is congruent to angle Q R S and angle R S P is congruent to angle P Q R

D.
angle S T P is congruent to angle Q T R and angle P T Q is congruent to angle R T S

Answer 1

Answer: A and C.

Reasoning:
- A: If ΔPQR ≅ ΔRSP then corresponding sides give PQ = RS and QR = PS, so both pairs of opposite sides are equal → PQRS is a parallelogram.
- C: The given equalities make opposite angles equal (∠P = ∠R and ∠Q = ∠S). Hence consecutive angles are supplementary, which implies each pair of opposite sides is parallel → PQRS is a parallelogram.

Why the others fail:
- B: PQ ∥ SR and PS = QR can occur in an isosceles trapezoid (non-parallelogram), so this is not sufficient.
- D: The equalities are simply pairs of vertical angles at the intersection of the diagonals (always true for any crossing diagonals) and do not force the figure to be a parallelogram.