Theorem on the Sum of Angles in a Convex Polygon

For a convex nn-gon, the sum of its interior angles is given by the formula:

Proof

To prove this theorem, we use the already established fact that the sum of the interior angles of a triangle is 180°.

Let A₁A₂...A_n be a given convex polygon with n > 3. 

We draw all diagonals from vertex A₁, which divide the polygon into n - 2 triangles:

Since the sum of the angles in each triangle is 180°, and there are n - 2 triangles, the total sum of the interior angles of the convex polygon is:

180°(n−2)

Problem 1: Number of Angles in a Convex Polygon

A convex polygon has three angles of 80°, and the rest are 150°. Find the total number of angles in the polygon.

Solution

Using the theorem on the sum of the interior angles:

where x is the number of unknown angles.
Since the total number of angles in the polygon is n, we express x as:

Substituting into the equation:

Expanding and solving for nn:

Answer: 5 angles

Problem 2: Number of Vertices in a Polygon

Find the possible number of vertices in a polygon if each interior angle is less than 120°.

Solution

Using the sum of interior angles formula:

Expanding:

Thus, if all angles were exactly 120°, the polygon would have 6 vertices.
If all angles are less than 120°, then nn must be less than 6.

Answer: Less than 6 vertices

Problem 3: Finding the Number of Vertices in a Polygon

A polygon has three angles of 113°, while the remaining angles are equal, and their measures are whole numbers.
Find the total number of vertices.

Solution

Using the theorem on the sum of exterior angles of a convex polygon:

The exterior angle corresponding to an interior angle is:

So, for the three known angles of 113°:

Since 159 can only be factored as 3×53, where 53 is a prime number, the only possible solution is:

Answer: 6 angles

Problem 4: Number of Acute Angles in a Convex Polygon

Prove that a convex polygon can have at most three acute angles.

Proof

We use the fact that the sum of the exterior angles of a convex polygon is always 360°.

Proof by contradiction:
Suppose the convex polygon has at least four acute interior angles.
Since each acute angle is less than 90°, the corresponding exterior angle is greater than 90°.

Thus, if there are at least four such angles, the total sum of exterior angles would be:

which contradicts the theorem that the total sum of all exterior angles is exactly 360°.

Thus, a convex polygon cannot have more than three acute angles.

Answer: No more than three acute angles