Geometry Rhombus Property Application
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The image shows a quadrilateral H-O-P-E. The angle $\angle OHP = 84^\circ$ and the angle $\angle OPE = 71^\circ$. Determine the measures of the other interior angles.
This question includes visual content: The image displays a quadrilateral with vertices labeled H, O, P, E in a diamond shape. The segments HO and HP are marked with single slashes, while EO and EP are marked with double slashes. A horizontal line segment HP (likely meant to be HO/HP or diagonals) connects the vertices. The angle at vertex H (specifically angle OHP) is labeled $84^\circ$. The angle at vertex P (specifically angle OPE) is labeled $71^\circ$. There are markings at angles O and E indicating they are bisected by the diagonal.
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Step by Step Written Solution
Hi Nichole, let's solve this geometry problem together by using the properties of a kite.
Using Kite Properties
From the diagram, we can see two pairs of adjacent equal sides marked with matching tick marks. That tells us the quadrilateral is a kite.
The vertical diagonal from H to P is the axis of symmetry of the kite. In a kite, this diagonal bisects the top and bottom angles.
Because the diagonal bisects the angles, each top half-angle is equal and each bottom half-angle is equal.
Now let's focus on the top angle at H.
Top Angle
Since the diagonal bisects the angle at H, the other half is also eighty eight degrees.
To find the whole angle at H, we add the two equal halves together.
Eighty eight plus eighty eight equals one hundred seventy six degrees.
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