7133 /www/nrich/html/content/id/7133/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <br></br> <p>In the diagram, $QSR$ is a straight line, $\angle QPS = 12^{\circ}$ and $PQ=PS=RS$.<br></br> What is the size of $\angle QPR$?<br></br> <br></br> <mdo:image alt="" height="234" src="39%202010.PNG" width="338"></mdo:image><br></br> <br></br> If you liked this problem, <a href="http://nrich.maths.org/1951&amp;part=">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br> <br></br>  </p> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Observing that triangle $PQS$ is isosceles, we have $\angle PSQ = \frac {1}{2}(180^{\circ} - 12^{\circ}) = 84^{\circ}$ and hence $\angle PSR = 180^{\circ} - 84^{\circ}=96^{\circ}$.<br></br>  <br></br> Since triagle $PRS$ is also isosceles, we have $\angle SPR = \frac {1}{2}(180^{\circ} - 96^{\circ}) = 42^{\circ}$. Hence $\angle QPR = 12^{\circ}+ 42^{\circ} = 54^{\circ}$.<br></br>  <br></br>  </p></mdoxml> 2 3 0 0 1 0 0 Weekly Problem 39 - 2010 2D Geometry, Shape and Space Angles 2D Geometry, Shape and Space Isosceles triangles 2D Geometry, Shape and Space Angle properties of shapes 2D Geometry, Shape and Space Angles at a point/on a line