# U in a Pentagon

Weekly Problem 18 - 2008

The diagram shows a regular pentagon. Can you work out the size of the marked angle?

The diagram shows a regular pentagon. Can you work out the size of the marked angle?

## Problem

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The diagram shows a regular pentagon $PQRST$. The lines $QS$ and $RT$ meet at $U$. What is the size of angle $PUR$?

If you liked this question, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

Each interior angle of a regular pentagon is $108^\circ$, so $\angle SRQ=108^\circ$. As $SR=QR$, the triangle is isosceles with $\angle RQS=\angle RSQ = 36^\circ$. Similarly, $\angle SRT= \angle STR = 36^\circ$. So $\angle SUR=(180-2\times36)^\circ=108^\circ$. From the symmetry of the figure, $\angle PUR=\angle PUS= (360^\circ - 108^\circ)/2 = 126^\circ$.