Interior angles of a convex polygon

A polygon is a closed plane figure made of straight sides. In a convex polygon every interior angle is less than \(180^\circ\). Drawing diagonals from one vertex divides a convex \(n\)-gon into \(n-2\) triangles.

Interior-angle sum \(=(n-2)\times180^\circ\). The angles need not be equal. For a regular polygon, all sides and all interior angles are equal, so each interior angle is \(\dfrac{(n-2)180^\circ}{n}\).

Example

A convex pentagon has angles \(95^\circ,110^\circ,120^\circ,105^\circ\) and \(x\). Its sum is \(3\times180^\circ=540^\circ\), so \(x=540^\circ-430^\circ=110^\circ\).

Equal sides alone do not make a polygon regular: a non-square rhombus is a counterexample.