An isosceles triangle PQR, in which PQ = PR, is split into three separate isosceles triangles, as shown, so that PS = SQ, RT = RS and QT = RT. What is the size, in degrees, of angle QPR?
A 24 B 28 C 30 D 35 E 36
E
Let the size, in degrees, of angle QPR be x. Since triangle PSQ is isosceles, angle PQS = x and, using the external angle theorem, angle RST = 2x. Since triangle ST R is isosceles, angle ST R = 2x and, since angles on a straight line add to 180◦ , angle QT R = 180−2x. Since triangle QT R is isosceles and angles in a triangle add to 180◦ , angle TQR = (180 − (180 − 2x))/2 = x. Therefore angle PQR = x + x = 2x and, since triangle PQR is also isosceles, angle PRQ = 2x. Therefore, in triangle PQR, we have x + 2x + 2x = 180, since angles in a triangle add to 180◦ . Hence x = 36 and so the size, in degrees, of angle QPR is 36.
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