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?
Age
11 to 14
Challenge level
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 $\angle QPR = 12^{\circ}+ 42^{\circ} = 54^{\circ}$.