Determine unknown interior angles of polygons

Interior-angle measures are related by the polygon angle-sum relationship $(n-2)\times180^\circ$ , derived by partitioning an $n$ -sided polygon into triangles. This understanding supports finding missing angles in irregular polygons and determining each angle in regular polygons, including cases expressed with simple equations, while distinguishing interior from exterior angles and avoiding the assumption that all angles are equal without evidence of regularity; the scope is limited to ordinary convex polygons rather than self-intersecting or more advanced generalized cases.

Detailed Explanation: Determine unknown interior angles of polygons

For any ordinary convex polygon with $n$ sides, the sum of its interior angles is

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

This works because a polygon can be divided into $(n-2)$ triangles, and each triangle has an angle sum of $(180^\circ)$ .

Worked example

A pentagon has interior angles measuring $(110^\circ)$ , $(95^\circ)$ , $(130^\circ)$ , $(115^\circ)$ , and $(x^\circ)$ . Find $x$ .

Step 1: Count the sides.

The shape is a pentagon, so $(n=5)$ .

Step 2: Find the total interior-angle sum.

(5-2)\times 180^\circ=3\times180^\circ=540^\circ

So, all five interior angles together measure $(540^\circ)$ .

Step 3: Add the known angles.

110^\circ+95^\circ+130^\circ+115^\circ=450^\circ

Step 4: Subtract from the total.

x=540^\circ-450^\circ

\boxed{x=90^\circ}

The unknown interior angle measures $\boxed{90^\circ}$ .

Remember: Use the interior angles inside the polygon. Do not assume the angles are equal unless the polygon is stated to be regular.

Learn by doing: Determine unknown interior angles of polygons

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Geometry Polygon Interior Angles - Polygon and Image to Missing Angle

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