CEMC Fermat, 2019

Geometry Difficulty 4.6 AIME Find the answer Canada

The five sides of a regular pentagon are all equal in length. Also, all interior angles of a regular pentagon have the same measure. In the diagram, $PQRST$ is a regular pentagon and $\triangle PUT$ is equilateral.

The measure of obtuse $\angle QUS$ is

Pick one

Solution

Join $QU$ and $SU$ .

Since $\triangle PUT$ is equilateral, then $PU=UT=TP$ .

Since pentagon $PQRST$ is regular, then $QP = PT = TS$ .

Thus, $PU = QP$ and $UT = TS$ , which means that $\triangle QPU$ and $\triangle STU$ are isosceles.

Each interior angle in a regular pentagon measures $108^\circ$ .

Since $\angle UPT = 60^\circ$ , then $\angle QPU = \angle QPT - \angle UPT = 108^\circ - 60^\circ = 48^\circ$ .

Since $\triangle QPU$ is isosceles with $QP = PU$ , then $\angle PQU = \angle PUQ$ .

Thus, $\angle PUQ = \frac{1}{2}(180^\circ - \angle QPU) = \frac{1}{2}(180^\circ - 48^\circ) = 66^\circ$ .

By symmetry, $\angle TUS = 66^\circ$

Finally, $\angle QUS = 360^\circ - \angle PUQ - \angle PUT - \angle TUS = 360^\circ - 66^\circ - 60^\circ - 66^\circ = 168^\circ$

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