Question

figure vwyx is a kite.
what is the measure of angle vxy?

• \\(72^\circ\\)
• \\(74^\circ\\)
• \\(106^\circ\\)
• \\(108^\circ\\)

Explanation:

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<reasoning>

Identify the properties of the kite

A kite is a quadrilateral with two pairs of equal adjacent sides. In the given kite \(VWYX\), we have:

• \(WV = WY\) (indicated by single tick marks)
• \(XV = XY\) (indicated by double tick marks)
• The interior angles at vertices \(V\) and \(Y\) are right angles, so \(\angle WVX = 90^\circ\) and \(\angle WYX = 90^\circ\).

Set up the angle sum equation

The sum of the interior angles of any quadrilateral is \(360^\circ\). For quadrilateral \(VWYX\):
\[
\angle W + \angle WVX + \angle X + \angle WYX = 360^\circ
\]
Substitute the given expressions and values:
\[
(18x - 2) + 90 + (12x + 2) + 90 = 360
\]

Solve for \(x\)

Combine the like terms in the equation:
\[
(18x + 12x) + (-2 + 2) + 90 + 90 = 360
\]
\[
30x + 180 = 360
\]
Subtract \(180\) from both sides:
\[
30x = 180
\]
Divide by \(30\):
\[
x = 6
\]

Calculate the measure of angle \(VXW\)

First, find the total measure of angle \(WXY\) at vertex \(X\):
\[
\angle WXY = 12x + 2 = 12(6) + 2 = 72 + 2 = 74^\circ
\]
In a kite, the diagonal connecting the vertices of the unequal angles (diagonal \(WX\)) bisects those angles. Therefore, the line segment \(WX\) bisects \(\angle WXY\).
The angle \(\angle VXY\) is the full angle at vertex \(X\), which is \(\angle WXY\).
Let's re-verify the question prompt: "What is the measure of angle VXY?"
Looking at the diagram, the vertex \(X\) has the interior angle labeled \((12x+2)^\circ\), which represents the entire angle \(\angle VXY\).
Thus, the measure of \(\angle VXY\) is:
\[
\angle VXY = 74^\circ
\]
</reasoning>

<answer>
<mcq-option>(A) \(72^\circ\)</mcq-option>
<mcq-correct>(B) \(74^\circ\)</mcq-correct>
<mcq-option>(C) \(106^\circ\)</mcq-option>
<mcq-option>(D) \(108^\circ\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the properties of the kite

A kite is a quadrilateral with two pairs of equal adjacent sides. In the given kite \(VWYX\), we have:

• \(WV = WY\) (indicated by single tick marks)
• \(XV = XY\) (indicated by double tick marks)
• The interior angles at vertices \(V\) and \(Y\) are right angles, so \(\angle WVX = 90^\circ\) and \(\angle WYX = 90^\circ\).

Set up the angle sum equation

The sum of the interior angles of any quadrilateral is \(360^\circ\). For quadrilateral \(VWYX\):
\[
\angle W + \angle WVX + \angle X + \angle WYX = 360^\circ
\]
Substitute the given expressions and values:
\[
(18x - 2) + 90 + (12x + 2) + 90 = 360
\]

Solve for \(x\)

Combine the like terms in the equation:
\[
(18x + 12x) + (-2 + 2) + 90 + 90 = 360
\]
\[
30x + 180 = 360
\]
Subtract \(180\) from both sides:
\[
30x = 180
\]
Divide by \(30\):
\[
x = 6
\]

Calculate the measure of angle \(VXW\)

First, find the total measure of angle \(WXY\) at vertex \(X\):
\[
\angle WXY = 12x + 2 = 12(6) + 2 = 72 + 2 = 74^\circ
\]
In a kite, the diagonal connecting the vertices of the unequal angles (diagonal \(WX\)) bisects those angles. Therefore, the line segment \(WX\) bisects \(\angle WXY\).
The angle \(\angle VXY\) is the full angle at vertex \(X\), which is \(\angle WXY\).
Let's re-verify the question prompt: "What is the measure of angle VXY?"
Looking at the diagram, the vertex \(X\) has the interior angle labeled \((12x+2)^\circ\), which represents the entire angle \(\angle VXY\).
Thus, the measure of \(\angle VXY\) is:
\[
\angle VXY = 74^\circ
\]
</reasoning>

<answer>
<mcq-option>(A) \(72^\circ\)</mcq-option>
<mcq-correct>(B) \(74^\circ\)</mcq-correct>
<mcq-option>(C) \(106^\circ\)</mcq-option>
<mcq-option>(D) \(108^\circ\)</mcq-option>
</answer>

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