Skill: Apply interior and exterior angle relationships for polygons

For an $n$-sided polygon, the sum of its interior angles is

$$(n-2)180^\circ.$$

If a side is extended at a vertex, the exterior angle and the adjacent interior angle form a straight line, so they are supplementary:

$$\text{interior angle}+\text{exterior angle}=180^\circ.$$

For a convex polygon, one exterior angle at each vertex always adds to

$$360^\circ.$$

Example

A regular octagon has $8$ equal sides. Find each interior angle and each exterior angle.

Step 1: Find the sum of the interior angles.

Use $(n=8)$:

$$(8-2)180^\circ=6(180^\circ)=1080^\circ.$$

Step 2: Divide by the number of angles.

Because the octagon is regular, all its interior angles are equal:

$$\frac{1080^\circ}{8}=135^\circ.$$

So, each interior angle is

$$\boxed{135^\circ}.$$

Step 3: Find each exterior angle.

An interior angle and its adjacent exterior angle are supplementary:

$$180^\circ-135^\circ=45^\circ.$$

So, each exterior angle is

$$\boxed{45^\circ}.$$

Check: There are $8$ equal exterior angles:

$$8(45^\circ)=360^\circ,$$

which matches the exterior-angle sum for a convex polygon.