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Geometry Difficulty 3.3 AMC 10/12 Find the answer Canada

In the diagram, $\$\triangle$
PQT $is equilateral. Both$ $\triangle$
QST $and$ $\triangle$ QRS$ are
right-angled triangles and $QR=RS$ .

If $\angle STP=120\degree$ , the
measure of $\angle PQR$ is

Pick one

Solution

Each angle in an equilateral triangle measures $60\degree$ , and so $\angle PQT=\angle QTP=60\degree$ . Since
$\angle STP=120\degree$ , then $\$\angle STQ=120\degree-\angle$ QTP=
$120\degree-60\degree=60\degree\$.$ The sum of the three angles in
$\triangle QST$ is $180\degree$ , and so $\$\angle TQS=180\degree-90\degree-60\degree=30\degree\$.$

In $\triangle QRS$ , $QR=RS$ and so $\$\angle QSR=\angle SQR=\dfrac{180\degree-90\degree}{2}=\dfrac{90\degree}{2}=45\degree\$.$

The measure of $\angle PQR$ is
equal to $\$\angle PQT+\angle TQS+\angle SQR=60\degree+30\degree+45\degree=135\degree\$.$

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