Angle Sums and Properties of Polygons

Why the angles in a triangle sum to 180°

Draw any triangle $ABC$, then draw a straight line through $A$ parallel to $BC$. The angle between $AB$ and this new line equals the angle at $B$, because alternate angles on parallel lines are equal. On the other side of $A$, another alternate angle equals the angle at $C$. These two angles and the original angle at $A$ lie along a straight line, so together they make $180°$. Therefore, for every triangle,

$A+B+C=180°.$

This is a proof, rather than a check using one particular drawing. For example, if two angles of a triangle are $43°$ and $68°$, the third is $180°-43°-68°=69°$.

From triangles to polygons

A polygon’s interior angles are its angles inside the shape. A quadrilateral can be split by a diagonal into two triangles, so its interior angles sum to $2\times180°=360°$. A convex pentagon can be split into three triangles by drawing diagonals from one vertex, giving $3\times180°=540°$.

For a convex polygon with $n$ sides, the same construction makes $n-2$ triangles. Hence its interior-angle sum is

$S=(n-2)\times180°.$

Other simple polygons can also be split into $n-2$ triangles, although for a concave polygon the diagonals may need to start from different vertices. The same angle-sum formula applies. A six-sided polygon, for instance, has an interior-angle sum of $(6-2)\times180°=720°$; its individual angles need not be equal.

Equal angles in a regular polygon

A regular polygon has equal sides and equal interior angles. Once you know the total, divide by the number of sides to find each interior angle:

$\text{each interior angle}=\frac{(n-2)\times180°}{n}.$

A regular pentagon therefore has five interior angles of $540°\div5=108°$ each. An exterior angle is formed by extending a side; it and the adjacent interior angle make a straight line. Each exterior angle of that pentagon is consequently $180°-108°=72°$.

Walking once around a polygon involves one full turn, so its exterior angles sum to $360°$. In a regular polygon they are equal, giving $\text{each exterior angle}=360°/n$. This also works in reverse: if each exterior angle is $30°$, the regular polygon has $360°\div30°=12$ sides.

Angle sums

Triangle: $A+B+C=180°$. One proof draws a line through a vertex parallel to the opposite side: alternate angles place all three angles on a straight line.

Polygon with $n$ sides: $S=(n-2)\times180°$, because it can be split into $n-2$ triangles.

Regular polygons

Each interior angle is $\dfrac{(n-2)\times180°}{n}$. Each exterior angle is $\dfrac{360°}{n}$, and an interior angle plus its adjacent exterior angle is $180°$.