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Geometry Difficulty 3.2 AMC 10/12 Find the answer Canada

In the diagram, $MNPQ$ is a rectangle with points $M$ , $N$ , $P$ , and $Q$ on the sides of $\triangle XYZ$ , as shown.

If $\angle{ZNM} = 68^\circ$ and $\angle{XYZ} = 55^\circ$ , what is the measure of $\angle{YXZ}$ ?

Pick one

Solution

Since $MNPQ$ is a rectangle, then $\angle NMQ$ is a right angle and thus so is $\angle NMZ$ .

The sum of the angles in $\triangle NMZ$ is $180^\circ$ , so $\angle NZM = 180^\circ - \angle ZNM - \angle ZMN = 180^\circ - 68^\circ - 90^\circ = 22^\circ$

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Since the sum of the angles in $\triangle ZXY$ is also $180^\circ$ , then $\angle YXZ = 180^\circ - \angle XZY - \angle XYZ = 180^\circ - 22^\circ - 55^\circ = 103^\circ$

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