Competition problem — Geometry — Maths Olympiad Prep

Library / /103 of 144

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. Fractions can be typed as 3/2, and spacing doesn't matter.

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 $\theta=36^{\circ}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.