Skill: 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.