14. In the diagram all the angles marked $\bullet$ are equal in size to the angle marked $x^{\circ}$.
What is the value of $x$ ?
A 100
B 105
C 110
D 115
E 120
SolUTION
A
Method 1
We label the vertices of the figure as shown below.
We place an arrow lying along $P X$ in the direction shown in the first figure on the left above. We rotate this arrow anticlockwise about the point $P$ until it lies along $P Q$ in the direction shown in the second figure above. The arrow has been turned anticlockwise through the angle $x^{\circ}$.

Next we rotate the arrow anticlockwise about the point $Q$ until it lies along $R Q$ in the direction shown in the third figure above. As all the angles marked $\bullet$ are $x^{\circ}$, the arrow has again been turned anticlockwise through $x^{\circ}$.
We continue this process, rotating the arrow anticlockwise through $x^{\circ}$ about the points $R, S, T$, $U, V, W$ and $X$ in turn. The arrow ends up lying along $X P$ in the direction shown in the figure on the right above.
In this figure we have also shown the direction in which the arrow points on all the other edges during this process, apart from its initial position.
It will be seen that in this process the arrow has been turned through $2 \frac{1}{2}$ complete revolutions. Therefore the total angle it has turned through is $2 \frac{1}{2} \times 360^{\circ}$, that is, through $900^{\circ}$. In the process the arrow been rotated 9 times through the angle $x^{\circ}$.
Therefore $9 x=900$ and hence $x=100$.
Method 2
We let $Y$ be the point where $R S$ meets $W X$. We also let $\angle R Y X=y^{\circ}$.

The sum of the angles of a pentagon is $540^{\circ}$. In the pentagon PXYRQ, the interior angle at $Y$ is $y^{\circ}$ and all the other four angles are $x^{\circ}$. Therefore we have
$$
4 x+y=540 .
$$

The sum of the angles of a hexagon is $720^{\circ}$. In the hexagon SYWVUT , the reflex angle at $Y$ is $360^{\circ}-y^{\circ}$ and all the other five angles are $x^{\circ}$.

Therefore $5 x+(360-y)=720$ and hence
$$
5 x-y=360 \text {. }
$$

By adding equations (1) and (2) we deduce that $9 x=900$. Therefore $x=100$.