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:

To solve this, we recall the properties of a kite: a kite has two distinct pairs of adjacent sides that are congruent, and one diagonal is the perpendicular bisector of the other (so it bisects the diagonal into two congruent segments), and one pair of opposite angles (the ones between the unequal sides) are congruent, and the angles between the congruent adjacent sides are congruent.

Analyzing each option:

1. $\boldsymbol{\overline{QP} \cong \overline{QR}}$: In a kite, two distinct pairs of adjacent sides are congruent. If $PQRS$ is a kite with $QP$ and $QR$ as adjacent sides of one pair, this could hold (adjacent sides congruent).
2. $\boldsymbol{\overline{PM} \cong \overline{MR}}$: The diagonal that connects the vertices between the non - congruent sides bisects the other diagonal. So if $PR$ is the diagonal bisected by the other diagonal, then $M$ (the intersection of the diagonals) divides $PR$ into two equal parts, so $\overline{PM} \cong \overline{MR}$ must be true.
3. $\boldsymbol{\overline{QR} \cong \overline{RS}}$: There's no guarantee that these are adjacent sides of a congruent pair. In a kite, only two distinct pairs of adjacent sides are congruent, so this is not necessarily true.
4. $\boldsymbol{\angle PQR \cong \angle PSR}$: In a kite, one pair of opposite angles (the ones between the sides that are not part of the congruent adjacent pairs) are congruent. $\angle PQR$ and $\angle PSR$ are such a pair, so this must be true.
5. $\boldsymbol{\angle QPS \cong \angle QRS}$: The angles between the congruent adjacent sides (the ones at the ends of the diagonal that is bisected) are congruent. So $\angle QPS$ and $\angle QRS$ (angles at $P$ and $R$ between the congruent adjacent sides) are congruent, so this must be true.

• $\overline{QP} \cong \overline{QR}$: Adjacent sides in a kite can be congruent (part of a pair of adjacent congruent sides).
• $\overline{PM} \cong \overline{MR}$: The diagonal bisected by the other diagonal is split into two congruent segments.
• $\angle PQR \cong \angle PSR$: One pair of opposite angles in a kite (between non - congruent side pairs) are congruent.
• $\angle QPS \cong \angle QRS$: Angles between congruent adjacent sides in a kite are congruent.
• $\overline{QR} \cong \overline{RS}$: Not a necessary property of a kite.

From the analysis, the three statements that must be true are $\overline{QP} \cong \overline{QR}$, $\overline{PM} \cong \overline{MR}$, $\angle PQR \cong \angle PSR$, $\angle QPS \cong \angle QRS$. But we need to select three. Wait, re - evaluating:

Wait, let's re - check the standard kite properties:

A kite has:

• Two pairs of adjacent sides congruent (e.g., $QP\cong QR$ and $SP\cong SR$ or $QP\cong QS$? No, standard definition: two distinct pairs of adjacent sides are congruent. So let's assume the kite has vertices in order $P, Q, R, S$, so sides $QP, QR$ (adjacent), $RS, SP$ (adjacent) as one possible pair, or $QP, PS$ and $QR, RS$ as the other.

The diagonal that is the axis of symmetry (the one that connects the "top" and "bottom" of the kite) bisects the other diagonal, so $PM = MR$.

The angles between the unequal sides (the ones not in the congruent adjacent pairs) are equal: $\angle PQR=\angle PSR$.

The angles at the ends of the bisected diagonal (the ones between the congruent adjacent sides) are equal: $\angle QPS=\angle QRS$.

And one pair of adjacent sides (like $QP$ and $QR$) can be congruent.

So the three true statements are $\overline{PM} \cong \overline{MR}$, $\angle PQR \cong \angle PSR$, $\angle QPS \cong \angle QRS$ (and $\overline{QP} \cong \overline{QR}$ can be true depending on the kite, but the three that must be true are $\overline{PM} \cong \overline{MR}$, $\angle PQR \cong \angle PSR$, $\angle QPS \cong \angle QRS$? Wait, no, let's correct:

Wait, the correct three are:

• $\overline{PM} \cong \overline{MR}$ (diagonal bisected),
• $\angle PQR \cong \angle PSR$ (opposite angles congruent),
• $\angle QPS \cong \angle QRS$ (angles between congruent adjacent sides congruent), and $\overline{QP} \cong \overline{QR}$ (adjacent sides congruent) – but we need to pick three. Wait, maybe the intended three are $\overline{QP} \cong \overline{QR}$, $\overline{PM} \cong \overline{MR}$, $\angle PQR \cong \angle PSR$? No, let's check again.

Wait, the standard kite properties:

1. Two distinct pairs of adjacent sides are congruent: So either $QP\cong QR$ and $SP\cong SR$ or $QP\cong PS$ and $QR\cong RS$. So $\overline{QP} \cong \overline{QR}$ can be true (if it's one of the adjacent pairs).

1. One diagonal is the perpendicular bisector of the other: So the diagonal (say $PR$) is bisected by the other diagonal, so $PM = MR$.

1. One pair of opposite angles are congruent: $\angle PQR \cong \angle PSR$.

1. The angles between the congruent adjacent sides are congruent: $\angle QPS \cong \angle QRS$.

So the three that must be true are $\overline{PM} \cong \overline{MR}$, $\angle PQR \cong \angle PSR$, $\angle QPS \cong \angle QRS$ (and $\overline{QP} \cong \overline{QR}$ is true if it's part of the adjacent pair, but the three that are always true are the ones related to diagonal bisecting and angle congruences). Wait, maybe the problem has a typo in the angle labels, but based on kite properties, the three correct options are:

$\overline{QP} \cong \overline{QR}$, $\overline{PM} \co…

Answer:

$\boldsymbol{\overline{QP} \cong \overline{QR}}$, $\boldsymbol{\overline{PM} \cong \overline{MR}}$, $\boldsymbol{\angle PQR \cong \angle PSR}$ (or $\boldsymbol{\angle QPS \cong \angle QRS}$ – depending on the kite's configuration, but the three valid options from the list are $\overline{QP} \cong \overline{QR}$, $\overline{PM} \cong \overline{MR}$, $\angle PQR \cong \angle PSR$, $\angle QPS \cong \angle QRS$; but we select three as per the question's "select three options" – the three that must be true are $\overline{PM} \cong \overline{MR}$, $\angle PQR \cong \angle PSR$, $\angle QPS \cong \angle QRS$ (and $\overline{QP} \cong \overline{QR}$ is also true if it's an adjacent pair). However, the most accurate three are:

• $\overline{QP} \cong \overline{QR}$ (adjacent sides congruent in a kite),
• $\overline{PM} \cong \overline{MR}$ (diagonal bisected),
• $\angle PQR \cong \angle PSR$ (opposite angles congruent in a kite),
• $\angle QPS \cong \angle QRS$ (angles between congruent adjacent sides congruent).

But since we need three, the three correct options are $\boldsymbol{\overline{QP} \cong \overline{QR}}$, $\boldsymbol{\overline{PM} \cong \overline{MR}}$, $\boldsymbol{\angle PQR \cong \angle PSR}$ (or $\boldsymbol{\angle QPS \cong \angle QRS}$).