Apply the interior angle-sum relationship for polygons

The interior angles of a simple convex polygon with $n$ sides sum to $(n-2) \times 180^\circ$ , a relationship explained by partitioning the polygon into $n-2$ triangles. This understanding supports finding unknown angles in triangles, quadrilaterals, and other familiar polygons, including determining each angle of a regular polygon and checking whether given measures are consistent; exterior-angle sums, self-intersecting polygons, and more advanced generalizations are outside this scope.

Detailed Explanation: Apply the interior angle-sum relationship for polygons

For a simple, convex polygon with $n$ sides, the sum of its interior angles is

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

This works because you can draw diagonals from one vertex to divide the polygon into $(n-2)$ triangles. Each triangle has angle sum $(180^\circ)$ .

Example

A pentagon has interior angles of $(110^\circ)$ , $(125^\circ)$ , $(95^\circ)$ , $(130^\circ)$ , and $(x^\circ)$ . Find $x$ .

Step 1: Count the sides.

A pentagon has $(n=5)$ sides.

Step 2: Find the total interior angle sum.

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

So, the five interior angles must add to $(540^\circ)$ .

Step 3: Add the known angles.

110^\circ+125^\circ+95^\circ+130^\circ=460^\circ

Step 4: Subtract from the total to find $x$ .

x=540^\circ-460^\circ=80^\circ

Therefore, the missing interior angle is

\boxed{80^\circ}.

To check, add all five angles:

110^\circ+125^\circ+95^\circ+130^\circ+80^\circ=540^\circ,

which matches the interior angle sum for a pentagon.

Learn by doing: Apply the interior angle-sum relationship for polygons

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Geometry Polygon Interior Angles - Polygon and Image to Missing Angle

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