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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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