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Problem 765

AMC 10/12, early questions

Geometry Difficulty 3.5 Multiple choice CEMC Fermat · Canada · 2026

In the diagram, point $P$ lies
inside square $WXYZ$ so that $\triangle PXY$ is an equilateral
triangle.

The measure of $\angle WPZ$ is

Pick one

Official solution

In equilateral $\triangle PXY$ , the measure of $\angle PXY$ is $60\degree$ and $PX=XY$ .

In square $WXYZ$ , the measure of
$\angle WXY$ is $90\degree$ and $WX=XY$ .

Therefore, $\angle WXP=\angle WXY-\angle PXY=90\degree-60\degree=30\degree$ and $WX=PX$ .

Thus $\triangle WXP$ is isosceles,
and so $\angle PWX=\angle WPX=\dfrac{180\degree-30\degree}{2}=75\degree$ .

The measure of $\angle PWZ$ is $90\degree-\angle PWX=90\degree-75\degree=15\degree$ .

In a similar way, we can show that $\angle PZW=15\degree$ , and so $\angle WPZ=180\degree-\angle PWZ-\angle PZW$ or $\angle WPZ=180\degree-15\degree-15\degree=150\degree$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.