## Angles of Polygons

This study guide focuses on polygons⁚ convex vs concave‚ interior and exterior angles‚ and basic terminology for common polygons. Polygon Exterior Angle The sum of the exterior angle measures of a convex polygon‚ one angle Sum Theorem at each vertex‚ is 360.

### Polygon Interior Angles Sum

The segments that connect the nonconsecutive vertices of a polygon are called diagonals. A diagonal divides a polygon into two smaller polygons. The sum of the measures of the interior angles of the polygon can be found by adding the measures of the interior angles of those n ‒ 2 triangles. For example‚ a quadrilateral can be divided into two triangles. Since the sum of the measures of the interior angles of a triangle is 180°‚ the sum of the measures of the interior angles of a quadrilateral is 2(180°) or 360°. In general‚ the sum of the measures of the interior angles of a polygon with n sides is given by the following formula⁚ Sum of Interior Angle Measures = (n ― 2)180°.

### Polygon Exterior Angle Sum Theorem

An exterior angle of a polygon is formed by extending one side of the polygon. The exterior angle and the interior angle at that vertex are supplementary. In the figure‚ ∠1 and ∠2 are supplementary. The sum of the measures of the exterior angles of a convex polygon‚ one angle at each vertex‚ is 360°. To prove this theorem‚ we can use the fact that the sum of the measures of the interior angles of a polygon with n sides is (n ― 2)180°. Since the sum of the measures of each interior and exterior angle is 180°‚ the sum of the measures of all interior and exterior angles is n(180°). Therefore‚ the sum of the measures of the exterior angles is n(180°) ‒ (n ‒ 2)180° = 360°.

### Finding the Sum of Interior Angles

The sum of the measures of the interior angles of a polygon can be found by adding the measures of the interior angles of those n ‒ 2 triangles. For example‚ a quadrilateral can be divided into two triangles‚ so the sum of the measures of its interior angles is 2(180°) = 360°. A pentagon can be divided into three triangles‚ so the sum of the measures of its interior angles is 3(180°) = 540°. In general‚ the sum of the measures of the interior angles of a polygon with n sides is (n ― 2)180°. This formula can be used to find the sum of the measures of the interior angles of any polygon‚ regardless of the number of sides. For example‚ the sum of the measures of the interior angles of a polygon with 10 sides is (10 ‒ 2)180° = 1440°.

### Finding the Measure of Each Interior Angle of a Regular Polygon

A regular polygon is a polygon with all sides congruent and all angles congruent. To find the measure of each interior angle of a regular polygon‚ you can use the formula (n ‒ 2)180°/n‚ where n is the number of sides. For example‚ a regular hexagon has 6 sides‚ so the measure of each interior angle is (6 ‒ 2)180°/6 = 120°. This formula can be used to find the measure of each interior angle of any regular polygon‚ regardless of the number of sides. For example‚ the measure of each interior angle of a regular polygon with 10 sides is (10 ― 2)180°/10 = 144°. This formula can be used to find the measure of each interior angle of any regular polygon‚ regardless of the number of sides.

## Examples

The following examples will demonstrate how to find the sum of the measures of the interior angles of a polygon‚ as well as the measure of each interior angle of a regular polygon.

### Example 1⁚ Finding the Sum of Interior Angles

Let’s say we have a convex polygon with 13 sides. To find the sum of the measures of its interior angles‚ we can use the formula (n ‒ 2) * 180‚ where ‘n’ represents the number of sides. In this case‚ ‘n’ is 13. So‚ the sum of the interior angles is (13 ‒ 2) * 180 = 11 * 180 = 1980 degrees.

This means that the sum of all the interior angles of a 13-sided polygon will always be 1980 degrees. This formula is a powerful tool for understanding the relationships between the number of sides of a polygon and the sum of its interior angles.

### Example 2⁚ Finding the Measure of Each Interior Angle of a Regular Polygon

Consider a regular hexagon‚ which has six equal sides and six equal angles. To find the measure of each interior angle‚ we can use the formula (n ― 2) * 180 / n‚ where ‘n’ is the number of sides. For a hexagon‚ ‘n’ is 6. Plugging this into the formula‚ we get (6 ― 2) * 180 / 6 = 4 * 180 / 6 = 120 degrees.

Therefore‚ each interior angle of a regular hexagon measures 120 degrees. This formula allows us to calculate the measure of each interior angle in any regular polygon‚ making it a valuable tool for understanding the properties of these geometric shapes.

## Practice Problems

Test your understanding of polygon angles with these practice problems.

### Practice Problem 1

A regular hexagon has six equal sides and six equal angles. Find the measure of each interior angle of the hexagon. Here’s how to solve it⁚

**Find the sum of the interior angles⁚**Use the formula (n ― 2) * 180‚ where n is the number of sides. For a hexagon (n = 6)‚ the sum is (6 ‒ 2) * 180 = 720 degrees.**Divide the sum by the number of angles⁚**Since a hexagon has six equal angles‚ divide the sum by 6⁚ 720 degrees / 6 = 120 degrees.

Therefore‚ each interior angle of a regular hexagon measures 120 degrees.

### Practice Problem 2

A regular pentagon has five equal sides and five equal angles. Determine the measure of each exterior angle of the pentagon. Here’s how to solve it⁚

**Remember the Exterior Angle Sum Theorem⁚**The sum of the measures of the exterior angles of any polygon is always 360 degrees.**Divide the sum by the number of angles⁚**Since a pentagon has five equal exterior angles‚ divide the sum by 5⁚ 360 degrees / 5 = 72 degrees.

Therefore‚ each exterior angle of a regular pentagon measures 72 degrees.

### Practice Problem 3

You are given a polygon with 10 sides. Your task is to find the sum of the measures of its interior angles. Here’s how to approach this problem⁚

**Recall the Polygon Interior Angles Sum Formula⁚**The sum of the interior angles of a polygon with *n* sides is given by (n ― 2) * 180 degrees.**Substitute the number of sides⁚**In this case‚ the polygon has 10 sides (n = 10). Substitute this value into the formula⁚ (10 ― 2) * 180 degrees = 8 * 180 degrees.**Calculate the sum⁚**8 * 180 degrees = 1440 degrees.

Therefore‚ the sum of the measures of the interior angles of a 10-sided polygon is 1440 degrees.

### Practice Problem 4

Imagine a regular polygon with each interior angle measuring 150 degrees. Your goal is to determine the number of sides this polygon has; Let’s break down the steps⁚

**Use the Formula for Interior Angle of a Regular Polygon⁚**The measure of each interior angle of a regular polygon with *n* sides is given by [(n ― 2) * 180 degrees] / n.**Set up an Equation⁚**We know the interior angle is 150 degrees. Substitute this into the formula⁚ 150 degrees = [(n ― 2) * 180 degrees] / n.**Solve for ‘n’⁚**- Multiply both sides by *n*⁚ 150n = (n ‒ 2) * 180.
- Expand the right side⁚ 150n = 180n ‒ 360.
- Subtract 180n from both sides⁚ -30n = -360.
- Divide both sides by -30⁚ n = 12.

Therefore‚ the regular polygon with each interior angle measuring 150 degrees has 12 sides.

### Practice Problem 5

Let’s consider a polygon with 10 sides. We want to find the sum of its interior angles. Here’s how to solve it⁚

**Recall the Formula⁚**The sum of the interior angles of a polygon with *n* sides is given by (n ‒ 2) * 180 degrees.**Substitute the Number of Sides⁚**In this case‚ n = 10. Substitute this into the formula⁚ (10 ‒ 2) * 180 degrees.**Calculate the Sum⁚**Simplify the expression⁚ (8) * 180 degrees = 1440 degrees.

Therefore‚ the sum of the interior angles of a 10-sided polygon is 1440 degrees.

Remember that this formula applies to all polygons‚ regardless of whether they are regular or irregular.