4. Tom draws a square. He adds a regular hexagon, one side of which joins the centre of the square to one of the vertices of the square, as shown. What is the size of angle $\alpha$ ?
A $105^{\circ}$
B $110^{\circ}$
C $115^{\circ}$
D $120^{\circ}$
E $125^{\circ}$
SolUtion
A

Let $T$ be the vertex of the requested angle. Look at the angles of the quadrilateral $O R Q T$.
$\angle O R Q=45^{\circ}$ as it is the angle between the side and diagonal of a square.
$\angle R Q T$ is a right angle as it is a corner of a square.
$\angle T O R=120^{\circ}$ as it is the interior angle of a hexagon.
Angles in a quadrilateral must add to $360^{\circ}$ hence:
$45^{\circ}+90^{\circ}+120^{\circ}+\alpha=360$ which gives $\alpha^{\circ}$ as $105^{\circ}$.