From the question it is given that, The ratio between an exterior angle and the interior angle of a regular polygon is 1: 5
Let us assume exterior angle be $$ y $$ And interior angle be $$ 5 y $$
We know that, sum of interior and exterior angle is equal to $$ 180^{\circ} $$,
$$\begin{array}{l}y+5 y=180^{\circ} \\6 y=180^{\circ} \\y=180^{\circ} / 6 \\y=30^{\circ}\end{array}$$
the number of sides in the polygon The number of sides of a regular polygon whose each interior angles has a measure of
$$150^{\circ}$$
Let us assume the number of sides of the regular polygon be $$ n $$,
Then, we know that $$ 150^{\circ}=((2 n-4) / n) \times 90^{\circ} $$
$$\begin{array}{l}150^{\circ} / 90^{\circ}=(2 n-4) / n \\5 / 3=(2 n-4) / n\end{array}$$
By cross multiplication,$$3(2 n-4)=5 n$$
$$6 n-12=5 n$$
By transposing we get,
$$\begin{array}{l}6 n-5 n=12 \\n=12\end{array}$$
Therefore, the number of sides of a regular polygon is 12 .