Apply interior and exterior angle relationships for polygons

The learner understands that the sum of the interior angles of an $n$ -sided polygon is $(n-2)180^\circ$ , and that an exterior angle formed by extending a side is supplementary to its adjacent interior angle; for a convex polygon, one exterior angle at each vertex sums to $360^\circ$ . This supports finding unknown angles and, for regular polygons, determining each interior or exterior angle from the number of sides, without extending to non-simple polygons or more advanced formal generalizations.

Detailed Explanation: 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.

Learn by doing: Apply interior and exterior angle relationships for polygons

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Sum of Inside Angles on a Shape

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