Question

5
in parallelogram wxyz, with diagonal xz, m\angle wxz = 81° and m\angle wzx = 28°. what is the measure of \angle xyz?
a. 90°
b. 62°
c. 71°
d. 109°
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Explanation:

Step1: Use the property of parallelogram

In a parallelogram \(WXYZ\), \(WX\parallel YZ\). So, \(\angle WXZ=\angle YZX\) (alternate - interior angles). Also, \(\angle WZX=\angle XZY\) (alternate - interior angles).
We know that in \(\triangle WXZ\), \(\angle W = 180^{\circ}-(\angle WXZ+\angle WZX)\) (angle - sum property of a triangle: \(\angle A+\angle B+\angle C = 180^{\circ}\)).
Since \(WX\parallel YZ\), \(\angle XYZ+\angle W=180^{\circ}\) (consecutive - interior angles of a parallelogram).
Another way:
We know that \(\angle WXY=\angle WXZ+\angle WZX\) (angle addition).
Since \(WX\parallel YZ\), \(\angle XYZ=\angle W\) (opposite angles of a parallelogram are equal).
First, find \(\angle W\) in \(\triangle WXZ\).
Using the angle - sum property of a triangle in \(\triangle WXZ\): \(\angle W=180^{\circ}-(81^{\circ}+28^{\circ})\)

Step2: Calculate \(\angle W\)

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Since \(WX\parallel YZ\) in parallelogram \(WXYZ\), \(\angle XYZ=\angle W\) (opposite angles of a parallelogram are equal)

Answer:

C. \(71^{\circ}\)