Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 2.3 Junior Find the answer

In triangle $XYZ$ , $XY=XZ$ and $W$ is on $XZ$ such that $XW=WY=YZ$ . What is the measure of $\angle XYW$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $\angle XYW=\theta$ . Since $\triangle XYW$ is isosceles with $WX=WY$ , then $\angle YXW=\angle XYW=\theta$ . Since the sum of the angles in $\triangle XYW$ is $180^{\circ}$ , then $\angle XWY=180^{\circ}-2\theta$ . Since $\angle XWY+\angle ZWY=180^{\circ}$ , then $\angle ZWY=180^{\circ}-(180^{\circ}-2\theta)=2\theta$ . Since $\triangle YWZ$ is isosceles with $YW=YZ$ , then $\angle YZW=\angle ZWY=2\theta$ . Since $\triangle XYZ$ is isosceles, with $XY=XZ$ , then $\angle XYZ=\angle XZY=2\theta$ . Since the sum of the angles in $\triangle XYZ$ is $180^{\circ}$ , then $\angle XYZ+\angle XZY+\angle YXZ=180^{\circ}$ or $2\theta+2\theta+\theta=180^{\circ}$ , or $5\theta=180^{\circ}$ , or $\theta=36^{\circ}$ .

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.