Converse Parallelogram Proofs Quick Check

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Question
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A parallelogram has vertices upper Q, upper R, upper S, and upper T. Opposite vertices are connected via diagonal lines that intersect at an interior point labeled upper P. Sides upper Q upper R and upper T upper S are much longer than the other sides.

It is given that quadrilateral QRST is a parallelogram. Which segment of the figure must be congruent to segment SP¯¯¯¯¯¯¯ ?

(1 point)
Responses

PR¯¯¯¯¯¯¯¯
upper P upper R

QS¯¯¯¯¯¯¯
upper Q upper S

QP¯¯¯¯¯¯¯¯
upper Q upper P

PT¯¯¯¯¯¯¯
upper P upper T
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Answer 1

In a parallelogram, opposite sides and diagonals have specific congruencies. Since quadrilateral QRST is given to be a parallelogram, we know that for any parallelogram:

1. Opposite sides are congruent: QR ≅ TS and QS ≅ RT
2. The diagonals bisect each other.

Since diagonal segments SR and QT share the point P at which they intersect, and diagonal segments SP and PT are parts of those diagonals, we can deduce that the segments must satisfy congruency as well.

Since P is the intersection point of the diagonals, the segments SP and PR are congruent as they are both halves of diagonal QT.

Thus, the segment that must be congruent to segment \( SP \) is:

PR (P R)