Angle Hunt
Age
11 to 14
Challenge level
Weekly Problem 39 - 2010
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 $\angle QPR = 12^{\circ}+ 42^{\circ} = 54^{\circ}$.