Inside a Regular Polygon

Many people think a diagonal always cuts a corner of a regular polygon into two equal halves. In a square it does. But the square is a special case.

A diagonal that skips one corner cuts off a triangle. Two sides of that triangle are sides of the polygon, so they are equal. A triangle with two equal sides is isosceles. Its two base angles, the angles facing the equal sides, are equal too.

In the square, the corner between the equal sides is 90°. So each base angle is $\frac{180-90}{2} = 45^{\circ}$.