Question

in the figure below, trapezoid wxyz is isosceles, with $overline{wz}congoverline{xy}$. if $mangle xwz = 125^{circ}$ and $mangle wxz = 27^{circ}$, what is $mangle xzy$? a. $27^{circ}$ b. $20^{circ}$ c. $55^{circ}$ d. $35^{circ}$

Explanation:

Step1: Recall properties of isosceles trapezoid

In an isosceles trapezoid \(WXYZ\) with \(WZ\cong XY\), \(\triangle WXZ\cong\triangle YXZ\) (by SSS - \(WZ = XY\), \(XZ=XZ\), \(WX = YX\) as base - angles are equal in isosceles trapezoid). So \(\angle WXZ=\angle YXZ = 27^{\circ}\) and \(\angle XWZ=\angle XYZ = 125^{\circ}\).

Step2: Use angle - sum property of triangle

In \(\triangle XYZ\), we know that the sum of interior angles of a triangle is \(180^{\circ}\). Let \(\angle XZY=\theta\). Then \(\angle XYZ+\angle YXZ+\theta = 180^{\circ}\).
We know \(\angle XYZ = 125^{\circ}\) and \(\angle YXZ = 27^{\circ}\).
Substitute the values into the formula: \(125^{\circ}+27^{\circ}+\theta=180^{\circ}\).
\(\theta=180^{\circ}-(125^{\circ} + 27^{\circ})\).
\(\theta = 180^{\circ}-152^{\circ}=28^{\circ}\) (There is a mistake above. Let's use another approach).
Since \(WXYZ\) is an isosceles trapezoid, \(WX\parallel ZY\). Then \(\angle WXZ=\angle XZY\) (alternate - interior angles). Given \(\angle WXZ = 27^{\circ}\), so \(\angle XZY=27^{\circ}\).

Answer:

A. \(27^{\circ}\)