Question

if quadrilateral pqrs is a kite, which statements must be true? select three options
□ $overline{qp} \cong overline{qr}$
□ $overline{pm} \cong overline{mr}$
□ $overline{qr} \cong overline{rs}$
□ $\angle pqr \cong \angle psr$
□ $\angle qps \cong \angle qrs$

Explanation:

1. Recall the properties of a kite: A kite has two distinct pairs of adjacent sides that are congruent. Also, the diagonal connecting the vertices between the unequal sides bisects the other diagonal and the angles at the vertices between the unequal sides.

• For $\overline{QP}\cong\overline{QR}$: In a kite, two adjacent sides (from the vertex where the unequal sides meet) can be congruent. If $QP$ and $QR$ are adjacent sides, this can hold.
• For $\overline{PM}\cong\overline{MR}$: The diagonal $QS$ (assuming the other diagonal is $PR$) bisects $PR$, so $PM = MR$, hence $\overline{PM}\cong\overline{MR}$.
• For $\angle PQR\cong\angle PSR$: In a kite, the angles between the unequal sides (the ones not between the congruent adjacent sides) are congruent. $\angle PQR$ and $\angle PSR$ are such angles.
• $\overline{QR}\cong\overline{RS}$: There is no guarantee that these are adjacent congruent sides (kite has two distinct pairs of adjacent congruent sides, not necessarily all sides in a way that $QR$ and $RS$ are congruent).
• $\angle QPS\cong\angle QRS$: These angles are not necessarily congruent based on kite properties.

1. So the three true statements are $\overline{QP}\cong\overline{QR}$, $\overline{PM}\cong\overline{MR}$, and $\angle PQR\cong\angle PSR$.

Answer:

• $\boldsymbol{\overline{QP}\cong\overline{QR}}$
• $\boldsymbol{\overline{PM}\cong\overline{MR}}$
• $\boldsymbol{\angle PQR\cong\angle PSR}$