CAT 2023 Slot 3 · Quant · Question 19

At each vertex the interior and exterior angles lie on a straight line, so they add to {180}^{∘}.

The interior angle exceeds the exterior by {120}^{∘}, so their difference is {120}^{∘}.

Adding the two relations, twice the interior angle is {300}^{∘}, so the interior angle is {150}^{∘}.

Subtracting, twice the exterior angle is {60}^{∘}, so the exterior angle is {30}^{∘}.

In any regular polygon the exterior angles sum to {360}^{∘}, so n=\frac{360}{30}=12.

A polygon with n vertices has \left( \genfrac{}{}{0pt}{}{n}{2}\right) lines joining pairs of vertices.

Of these, n are the sides themselves, so the diagonals number \left( \genfrac{}{}{0pt}{}{n}{2}\right)-n=\frac{n\left( n-3 \right)}{2}.

Substituting n=12, the count is \frac{12\times 9}{2}=54.

Shortcut. Interior and exterior angles differing by {120}^{∘} while summing to {180}^{∘} gives {150}^{∘} and {30}^{∘} immediately, and \frac{360}{30}=12 sides follows. The diagonal count \frac{n\left( n-3 \right)}{2} is worth memorising outright.