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

Question

Accepted Answer

# Explanation:
To solve this, we recall the properties of a kite: a kite has two distinct pairs of adjacent sides that are equal, one diagonal is the perpendicular bisector of the other, and the angles between the unequal sides are equal.

## Step 1: Analyze $ \overline{QP} \cong \overline{QR} $
In a kite, two adjacent sides are equal. Here, $ QP $ and $ QR $ are adjacent sides from $ Q $, so $ \overline{QP} \cong \overline{QR} $ is true.

## Step 2: Analyze $ \overline{PM} \cong \overline{MR} $
The diagonal $ QS $ is the perpendicular bisector of $ PR $, so it divides $ PR $ into two equal parts. Thus, $ \overline{PM} \cong \overline{MR} $ is true.

## Step 3: Analyze $ \overline{QR} \cong \overline{RS} $
In a kite, adjacent sides are equal. $ QR $ and $ RS $ are adjacent from $ R $, so $ \overline{QR} \cong \overline{RS} $ is true.

## Step 4: Analyze $ \angle PQR \cong \angle PSR $
In a kite, the angles between the unequal sides (the ones not between the equal adjacent sides) are not necessarily equal. So this is false.

## Step 5: Analyze $ \angle QPS \cong \angle QRS $
Using the properties of a kite (congruent triangles formed by diagonals), we can show $ \triangle QPS \cong \triangle QRS $ (by SSS, as $ QP = QR $, $ PS = RS $, $ QS $ common), so $ \angle QPS \cong \angle QRS $ is true? Wait, no, wait. Wait, earlier analysis: Wait, the correct three? Wait, let's recheck. Wait, the kite has two pairs of adjacent sides equal: $ QP = QR $ and $ SP = SR $. The diagonal $ QS $ is the axis of symmetry. So $ PM = MR $ (since $ QS $ bisects $ PR $). $ QR = RS $? Wait, no, $ QR $ and $ RS $: if $ QP = QR $ and $ SP = SR $, then $ QR = QP $, $ RS = SP $, but $ QP $ and $ SP $ may not be equal. Wait, maybe I made a mistake. Wait, the standard kite has two distinct pairs of adjacent congruent sides. So $ QP \cong QR $ (one pair), $ SP \cong SR $ (another pair). Then the diagonal $ QS $ is the perpendicular bisector of $ PR $, so $ PM \cong MR $. Then, triangles $ QPS $ and $ QRS $: $ QP = QR $, $ SP = SR $, $ QS = QS $, so by SSS, they are congruent. Thus, $ \angle QPS \cong \angle QRS $. Also, $ QR \cong RS $? Wait, no, $ QR $ is equal to $ QP $, $ RS $ is equal to $ SP $. Unless $ QP = SP $, which is not given. Wait, maybe the options: let's re-express.

Wait, the options:

1. $ \overline{QP} \cong \overline{QR} $: True (adjacent sides in kite)
2. $ \overline{PM} \cong \overline{MR} $: True (diagonal bisects the other)
3. $ \overline{QR} \cong \overline{RS} $: If $ QR $ and $ RS $ are adjacent, then yes (since kite has two pairs of adjacent congruent sides: $ QP=QR $ and $ SP=SR $, so $ QR=QP $, $ RS=SP $, but if $ QP=SP $, but no, the kite can have $ QP
eq SP $. Wait, maybe the diagram: in the diagram, $ PQRS $ is a kite with $ Q $ and $ S $ as the top and bottom, $ P $ and $ R $ on the left and right. So the two pairs of adjacent sides are $ QP=QR $ and $ SP=SR $. So $ QR \cong RS $? No, $ QR $ is adjacent to $ Q $ and $ R $, $ RS $ is adjacent to $ R $ and $ S $. So $ QR $ and $ RS $ are not a pair of adjacent congruent sides. Wait, maybe I messed up. Wait, the correct three: Let's check each:

- $ \overline{QP} \cong \overline{QR} $: True (adjacent sides)
- $ \overline{PM} \cong \overline{MR} $: True (diagonal $ QS $ bisects $ PR $)
- $ \overline{QR} \cong \overline{RS} $: False (unless $ QR=RS $, but not necessary)
- $ \angle PQR \cong \angle PSR $: False (angles between different sides)
- $ \angle QPS \cong \angle QRS $: True (from congruent triangles $ QPS $ and $ QRS $)

Wait, but the problem says "select three options". Wait, maybe my initial analysis is wrong. Wait, let's recall the kite properties:

- Two distinct pairs of adjacent sides are congruent.
- One diagonal is the perpendicular bisector of the other (so $ PM = MR $ if $ QS $ is the bisector).
- The angles between the unequal sides (the ones not in the pair) are equal? Wait, no, the angles between the congruent adjacent sides: the vertex angles between the two pairs of congruent sides are equal? Wait, no, in a kite, one pair of opposite angles (the ones between the unequal sides) are equal. Wait, maybe I confused.

Wait, let's use the diagram: $ PQRS $ is a kite with diagonals intersecting at $ M $. So $ QS \perp PR $ and $ PM = MR $ (since one diagonal bisects the other). Then, $ QP = QR $ (adjacent sides), $ SP = SR $ (adjacent sides). Then, triangles $ QPM $ and $ QRM $: $ QP = QR $, $ PM = MR $, $ QM = QM $, so congruent. Similarly, $ SPM $ and $ SRM $ congruent. Then, $ \angle QPS = \angle QRS $ (since $ \triangle QPS \cong \triangle QRS $ by SSS: $ QP=QR $, $ SP=SR $, $ QS=QS $). Also, $ \overline{QP} \cong \overline{QR} $ (true), $ \overline{PM} \cong \overline{MR} $ (true), $ \angle QPS \cong \angle QRS $ (true). Wait, but the option $ \overline{QR} \cong \overline{RS} $: is that true? $ QR $ is equal to $ QP $, $ RS $ is equal to $ SP $. Unless $ QP = SP $, which is not given. So maybe the three correct are $ \overline{QP} \cong \overline{QR} $, $ \overline{PM} \cong \overline{MR} $, and $ \angle QPS \cong \angle QRS $. But wait, the option $ \overline{QR} \cong \overline{RS} $: maybe in the diagram, $ QR = RS $? Wait, the diagram shows a kite that looks like a rhombus? No, a kite can be a rhombus, but not necessarily. Wait, maybe the problem's kite has $ QP=QR $ and $ QR=RS $? No, that would make it a rhombus. Wait, maybe I made a mistake. Let's check the options again:

Options:

1. $ \overline{QP} \cong \overline{QR} $: True (adjacent sides in kite)
2. $ \overline{PM} \cong \overline{MR} $: True (diagonal bisects the other)
3. $ \overline{QR} \cong \overline{RS} $: If $ QR $ and $ RS $ are adjacent, then yes (since kite has two pairs of adjacent congruent sides: $ QP=QR $ and $ SP=SR $, so $ QR=QP $, $ RS=SP $, but if $ QP=SP $, but no, the kite can have $ QP
eq SP $. Wait, maybe the diagram is a kite with $ QP=QR $ and $ SR=QR $? No, that would be three sides equal. Wait, maybe the correct three are $ \overline{QP} \cong \overline{QR} $, $ \overline{PM} \cong \overline{MR} $, and $ \overline{QR} \cong \overline{RS} $? No, that doesn't make sense. Wait, let's check the angle options:

$ \angle PQR \cong \angle PSR $: Are these angles equal? In a kite, the angles between the unequal sides (the ones not in the pair) are equal. Wait, $ \angle PQR $ is between $ QP $ and $ QR $, $ \angle PSR $ is between $ SP $ and $ SR $. Since $ QP=QR $ and $ SP=SR $, and $ QS $ is common, maybe those angles are equal? Wait, no, $ \angle PQR $ and $ \angle PSR $: let's see, in $ \triangle PQR $ and $ \triangle PSR $, $ QP=SP $? No, $ QP=QR $, $ SP=SR $. So unless $ QP=SP $, the triangles aren't congruent. So $ \angle PQR \cong \angle PSR $ is false.

$ \angle QPS \cong \angle QRS $: As $ \triangle QPS \cong \triangle QRS $ (SSS: $ QP=QR $, $ SP=SR $, $ QS=QS $), so this angle is equal. So true.

So the three true statements are:

- $ \overline{QP} \cong \overline{QR} $
- $ \overline{PM} \cong \overline{MR} $
- $ \angle QPS \cong \angle QRS $

Wait, but the option $ \overline{QR} \cong \overline{RS} $: maybe in the problem's kite, $ QR=RS $? Let me re-examine the diagram. The diagram shows $ PQRS $ with $ Q $ at top, $ S $ at bottom, $ P $ left, $ R $ right. The diagonals intersect at $ M $. So $ QP $ and $ QR $ are the left and right sides from $ Q $, $ SP $ and $ SR $ are the left and right sides from $ S $. So $ QP=QR $ (top two sides), $ SP=SR $ (bottom two sides). Then $ PM=MR $ (diagonal bisects the other). Then $ QR=RS $? $ QR $ is top-right, $ RS $ is bottom-right. So unless $ QR=RS $, which would mean $ QP=SP $, making it a rhombus. But the problem says it's a kite, not necessarily a rhombus. So maybe the correct three are $ \overline{QP} \cong \overline{QR} $, $ \overline{PM} \cong \overline{MR} $, and $ \overline{QR} \cong \overline{RS} $ is false, $ \angle QPS \cong \angle QRS $ is true. Wait, but the problem says "select three options". Maybe I made a mistake. Let's check standard kite properties:

1. Two distinct pairs of adjacent sides are congruent: $ QP \cong QR $, $ SP \cong SR $ (so first option true, third option: $ QR \cong RS $? If $ QR $ and $ RS $ are adjacent, then yes, because $ SP \cong SR $ and $ QP \cong QR $, but $ QR $ and $ RS $ are not a pair. Wait, no, the two pairs are $ (QP, QR) $ and $ (SP, SR) $. So $ QR $ is in the first pair, $ RS $ is in the second pair. So $ QR \cong RS $ would mean $ QP \cong SP $, which is not necessary. So maybe the three correct are:

- $ \overline{QP} \cong \overline{QR} $ (true)
- $ \overline{PM} \cong \overline{MR} $ (true)
- $ \angle QPS \cong \angle QRS $ (true)

But let's check the options again. The options are:

1. $ \overline{QP} \cong \overline{QR} $ – True
2. $ \overline{PM} \cong \overline{MR} $ – True
3. $ \overline{QR} \cong \overline{RS} $ – Maybe True? If the kite has $ QR=RS $, but not necessarily. Wait, maybe the diagram is a kite with $ QP=QR $ and $ QR=RS $, making it a rhombus. But a rhombus is a type of kite. So if it's a rhombus, then $ QR=RS $. But the problem says "kite", which includes rhombuses. So maybe in this case, $ QR \cong RS $ is true. Then the three would be 1, 2, 3? But then $ \angle QPS \cong \angle QRS $ would also be true. Wait, the problem says "select three options". Let's check each:

- $ \overline{QP} \cong \overline{QR} $: True (adjacent sides)
- $ \overline{PM} \cong \overline{MR} $: True (diagonal bisects the other)
- $ \overline{QR} \cong \overline{RS} $: True (if it's a rhombus, which is a kite)
- $ \angle PQR \cong \angle PSR $: False (angles not equal)
- $ \angle QPS \cong \angle QRS $: True (from congruent triangles)

But the problem says "select three options". So maybe the intended answers are $ \overline{QP} \cong \overline{QR} $, $ \overline{PM} \cong \overline{MR} $, and $ \overline{QR} \cong \overline{RS} $, or $ \overline{QP} \cong \overline{QR} $, $ \overline{PM} \cong \overline{MR} $, and $ \angle QPS \cong \angle QRS $. Wait, let's check with the kite definition. A kite has two pairs of adjacent congruent sides. So $ QP \cong QR $ (one pair), $ SP \cong SR $ (another pair). Then, the diagonal $ QS $ is the perpendicular bisector of $ PR $, so $ PM \cong MR $. Then, triangles $ QPS $ and $ QRS $ are congruent (SSS: $ QP=QR $, $ SP=SR $, $ QS=QS $), so $ \angle QPS \cong \angle QRS $. Also, $ QR \cong RS $ would mean $ QP \cong SP $, which is not necessary. So the three true are:

1. $ \overline{QP} \cong \overline{QR} $
2. $ \overline{PM} \cong \overline{MR} $
3. $ \angle QPS \cong \angle QRS $

But let's check the options again. The options are:

- $ \overline{QP} \cong \overline{QR} $ – True
- $ \overline{PM} \cong \overline{MR} $ – True
- $ \overline{QR} \cong \overline{RS} $ – Maybe True (if it's a rhombus)
- $ \angle PQR \cong \angle PSR $ – False
- $ \angle QPS \cong \angle QRS $ – True

So the three correct are $ \overline{QP} \cong \overline{QR} $, $ \overline{PM} \cong \overline{MR} $, and $ \angle QPS \cong \angle QRS $, or $ \overline{QP} \cong \overline{QR} $, $ \overline{PM} \cong \overline{MR} $, and $ \overline{QR} \cong \overline{RS} $. But according to standard kite properties, the first two and the angle one are correct.

# Answer:
The three true statements are:
- $ \overline{QP} \cong \overline{QR} $
- $ \overline{PM} \cong \overline{MR} $
- $ \angle QPS \cong \angle QRS $

(Note: If the kite is a rhombus, $ \overline{QR} \cong \overline{RS} $ would also be true, but the most general kite properties confirm the three above.)

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QUESTION IMAGE

Question

Explanation:

Step 1: Analyze \( \overline{QP} \cong \overline{QR} \)

Step 2: Analyze \( \overline{PM} \cong \overline{MR} \)

Step 3: Analyze \( \overline{QR} \cong \overline{RS} \)

Step 4: Analyze \( \angle PQR \cong \angle PSR \)

Step 5: Analyze \( \angle QPS \cong \angle QRS \)

Answer: