24. An isosceles triangle $P Q R$, in which $P Q=P R$, is split into three separate isosceles triangles, as shown, so that $P S=S Q, R T=R S$ and $Q T=R T$.
What is the size, in degrees, of angle $Q P R$ ?
A 24
B 28
C 30
D 35
E 36
SolUtion
Let the size, in degrees, of angle $Q P R$ be $x$. Since triangle $P S Q$ is isosceles, angle $P Q S=x$ and, using the external angle theorem, angle $R S T=2 x$. Since triangle $S T R$ is isosceles, angle $S T R=2 x$ and, since angles on a straight line add to $180^{\circ}$, angle $Q T R=180-2 x$. Since triangle $Q T R$ is isosceles and angles in a triangle add to $180^{\circ}$, angle $T Q R=(180-(180-2 x)) / 2=x$. Therefore angle $P Q R=x+x=2 x$ and, since triangle $P Q R$ is also isosceles, angle $P R Q=2 x$. Therefore, in triangle $P Q R$, we have $x+2 x+2 x=180$, since angles in a triangle add to $180^{\circ}$. Hence $x=36$ and so the size, in degrees, of angle $Q P R$ is 36 .