# Angle Hunt

Weekly Problem 39 - 2010

If you know three lengths and an angle in this diagram, can you find another angle by calculation?

If you know three lengths and an angle in this diagram, can you find another angle by calculation?

## Problem

In the diagram, $QSR$ is a straight line, $\angle QPS = 12^{\circ}$ and $PQ=PS=RS$.

What is the size of $\angle QPR$?

Image

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

## Student Solutions

Observing that triangle $PQS$ is isosceles, we have $\angle PSQ = \frac {1}{2}(180^{\circ} - 12^{\circ}) = 84^{\circ}$ and hence $\angle PSR = 180^{\circ} - 84^{\circ}=96^{\circ}$.

Since triagle $PRS$ is also isosceles, we have $\angle SPR = \frac {1}{2}(180^{\circ} - 96^{\circ}) = 42^{\circ}$. Hence $\angle QPR = 12^{\circ}+ 42^{\circ} = 54^{\circ}$.