Answer :

To prove that quadrilateral PQRS is a parallelogram, let's analyze the given conditions and use properties of lines and triangles. We are provided with the following information in the problem:

  1. PQ is parallel to SP (PQ || SP).
  2. The diagonals intersect at O and QO = OS.

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. We need to show that this condition holds for quadrilateral PQRS.

Steps to prove PQRS is a parallelogram:

  1. Parallel Condition: The problem states PQ || SP. Therefore, opposite sides PQ and SP are already parallel. We must now show that the other pair of opposite sides, QR and PS, are also parallel.

  2. Diagonals and Midpoints: The diagonals of PQRS intersect at point O where QO = OS. This indicates that O is the midpoint of both diagonals QS and PR because having QO = OS implies that O divides QS into two equal parts.

  3. Midsegment Theorem: In triangle QOS, PO || QS and O is the midpoint of QS. In triangle PQR, we can similarly argue that R is the midpoint of QP if OP = OR. Therefore, defining each diagonal to have the same midpoint implies that diagonals bisect each other.

  4. Conclusion of Parallel Sides: By the properties of diagonals bisecting each other in a quadrilateral, we conclude that both pairs of opposite sides are parallel:

    • Since PQ || SP was given, and we’ve shown that each diagonal bisects, the lines QR || PS due to congruent opposite sides formed by bisecting diagonals.

Therefore, we have:

  • PQ || SP
  • QR || PS

Thus, quadrilateral PQRS is a parallelogram by definition, since both pairs of opposite sides are parallel.

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