Converse Parallelogram Proofs Quick Check

Question

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Question
Use the image to answer the question.

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

QS¯¯¯¯¯¯¯
upper Q upper S

PT¯¯¯¯¯¯¯
upper P upper T

QP¯¯¯¯¯¯¯¯
upper Q upper P

PR¯¯¯¯¯¯¯¯
upper P upper R
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Answer 1

In a parallelogram, opposite sides are congruent, and the diagonals bisect each other.

Given that quadrilateral QRST is a parallelogram and that segment SP is one of the diagonals intersecting at point P, we know that point P is the midpoint of both diagonals \( QS \) and \( RT \). Consequently, segments \( SP \) and \( PT \) are congruent because \( P \) is the midpoint of diagonal \( QT \).

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

PT (segment \( P \) to \( T \)).