CEMC Fermat

Geometry Difficulty 2.7 Junior Find the answer

In an equilateral triangle $\triangle PRS$ , if $QS=QT$ and $\angle QTS=40^\circ$ , what is the value of $x$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $\triangle PRS$ is equilateral, then all three of its angles equal $60^\circ$ . In particular, $\angle RSP=60^\circ$ . Since $QS=QT$ , then $\triangle QST$ is isosceles and so $\angle TSQ=\angle STQ=40^\circ$ . Since $RST$ is a straight line segment, then $\angle RSP+\angle PSQ+\angle TSQ=180^\circ$ . Therefore, $60^\circ+x^\circ+40^\circ=180^\circ$ or $x=180-60-40=80$ .

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