Geometry (A): READ: Interior Angles Reading

Interior Angles

Learning Objectives

Identify the interior angles of convex polygons.
Find the sums of interior angles in convex polygons.
Identify the special properties of interior angles in convex quadrilaterals.

Introduction

By this point, you have studied the basics of geometry and you’ve spent some time working with triangles. Now you will begin to see some ways to apply your geometric knowledge to other polygons. This chapter focuses on quadrilaterals—polygons with four sides.

Note: Throughout this chapter, any time we talk about polygons, we will assume that we are talking about convex polygons.

Interior Angles in Convex Polygons

The interior angles are the angles on the inside of a polygon.

As you can see in the image, a polygon has the same number of interior angles as it does sides.

Summing Interior Angles in Convex Polygons

You have already learned the Triangle Sum Theorem. It states that the sum of the measures of the interior angles in a triangle will always be $180^\circ$ . What about other polygons? Do they have a similar rule?

We can use the triangle sum theorem to find the sum of the measures of the angles for any polygon. The first step is to cut the polygon into triangles by drawing diagonals from one vertex. When doing this you must make sure none of the triangles overlap.

Notice that the hexagon above is divided into four triangles.

Since each triangle has internal angles that sum to $180^\circ$ , you can find out the sum of the interior angles in the hexagon. The measure of each angle in the hexagon is a sum of angles from the triangles. Since none of the triangles overlap, we can obtain the TOTAL measure of interior angles in the hexagon by summing all of the triangles' interior angles. Or, multiply the number of triangles by $180^\circ$ :

$4(180^\circ) = 720^\circ$

The sum of the interior angles in the hexagon is $720^\circ$ .

Example 1

What is the sum of the interior angles in the polygon below?

The shape in the diagram is an octagon. Draw triangles on the interior using the same process.

The octagon can be divided into six triangles. So, the sum of the internal angles will be equal to the sum of the angles in the six triangles.

$6(180^\circ) = 1080^\circ$

So, the sum of the interior angles is $1080^\circ$ .

What you may have noticed from these examples is that for any polygon, the number of triangles you can draw will be two less than the number of sides (or the number of vertices). So, you can create an expression for the sum of the interior angles of any polygon using $n$ for the number of sides on the polygon.

The sum of the interior angles of a polygon with $n$ sides is

$\text{Angle Sum} = 180^\circ\ (n - 2).$

Example 2

What is the sum of the interior angles of a nonagon?

To find the sum of the interior angles in a nonagon, use the expression above. Remember that a nonagon has nine sides, so $n$ will be equal to nine.

$\text{Angle sum}&=180^\circ(n-2)\\ &=180^\circ(9-2) \\ &=180^\circ(7) \\ &=1260^\circ$

So, the sum of the interior angles in a nonagon is $1260^\circ$ .

Interior Angles in Quadrilaterals

A quadrilateral is a polygon with four sides, so you can find out the sum of the interior angles of a convex quadrilateral using our formula.

Example 3

What is the sum of the interior angles in a quadrilateral?

Use the expression to find the value of the interior angles in a quadrilateral. Since a quadrilateral has four sides, the value of $n$ will be $4$ .

$\text{Angle Sum} &= 180^\circ(n-2) \\ &= 180^\circ(4-2)\\ &= 180^\circ(2) \\ &= 360^\circ$

So, the sum of the measures of the interior angles in a quadrilateral is $360^\circ$ .

This will be true for any type of convex quadrilateral. You’ll explore more types later in this chapter, but they will all have interior angles that sum to $360^\circ$ . Similarly, you can divide any quadrilateral into two triangles. This will be helpful for many different types of proofs as well.

Lesson Summary

In this lesson, we explored interior angles in polygons. Specifically, we have learned:

How to identify the interior angles of convex polygons.
How to find the sums of interior angles in convex polygons.
How to identify the special properties of interior angles in convex quadrilaterals.

Understanding the angles formed on the inside of polygons is one of the first steps to understanding shapes and figures. Think about how you can apply what you have learned to different problems as you approach them.

The following questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

Copy the polygon below and show how it can be divided into triangles from one vertex.
Using the triangle sum theorem, what is the sum of the interior angles in this pentagon?

3-4: Find the sum of the interior angles of each polygon below.

Number of sides =

Sum of interior angles =

Number of sides =

Sum of interior angles =

Complete the following table:

Polygon name	Number of sides	Sum of measures of interior angles
triangle	$4$
	$5$
	$6$
	$7$
octagon
decagon
		$1,800^\circ$
	$n$

A regular polygon is a polygon with congruent sides and congruent angles. What is the measure of each angle in a regular pentagon?
What is the measure of each angle in a regular octagon?
Can you generalize your answer from 6 and 7? What is the measure of each angle in a regular $n-$ gon?
Can you use the polygon angle sum theorem on a convex polygon? Why or why not? Use the convex quadrilateral $ABCD$ to explain your answer.
If we know the sum of the angles in a polygon is $2700^\circ$ , how many sides does the polygon have? Show the work leading to your answer.

Review Answers

One possible answer:
$3(180) = 540^\circ$
Number of sides $= 7$ , sum of interior angles $= 900^\circ$
Number of sides $= 6$ , sum of interior angles $= 720^\circ$

Polygon name	Number of sides	Sum of measures of interior angles
triangle	$3$	$180^\circ$
quadrilateral	$4$	$360^\circ$
pentagon	$5$	$540^\circ$
hexagon	$6$	$720^\circ$
heptagon	$7$	$900^\circ$
octagon	$8$	$1,080^\circ$
decagon	$10$	$1,440^\circ$
dodecagon	$12$	$1,800^\circ$
$n-$ gon	$n$	$180 (n - 2)^\circ$

Since the sum of the angles is $540$ , each angle measures $\frac{540} {5} = 108^\circ$
$\frac{1080} {8} = 135^\circ$
$\frac{180(n - 2)} {n}$
Answers will vary. One possibility is no, we cannot use the polygon angle sum theorem because $\angle{C}$ is an acute angle that does not open inside the polygon. Alternatively, if we allow for angles between $180^\circ$ and $360^\circ$ , then we can use the angle sum theorem, but so far we have not seen angles measuring more than $180^\circ$
Solve the equation:
$180(n - 2) & = 2700 \\ \frac{180(n - 2)} {180} & = \frac{2700} {180} \\ n - 2 &= 15 \\ n - 2 + 2 &= 15 + 2 \\ n &= 17$

Last modified: Monday, June 28, 2010, 4:08 PM