# 6.1: Angles in Polygons

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Extend the concept of interior and exterior angles from triangles to convex polygons.
• Find the sums of interior angles in convex polygons.

## Review Queue

1. What do the angles in a triangle add up to?
2. Find the measure of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
3. A linear pair adds up to _____.
1. Find \begin{align*}w^\circ, x^\circ, y^\circ\end{align*}, and \begin{align*}z^\circ\end{align*}.
2. What is \begin{align*}w^\circ + y^\circ + z^\circ\end{align*}?

Know What? In nature, geometry is all around us. For example, sea stars have geometric symmetry. The common sea star, top, has five arms, but some species have over 20! To the right are two different kinds of sea stars. Name the polygon that is created by joining their arms and determine if either polygon is regular.

## Interior Angles in Convex Polygons

In Chapter 4, you learned that interior angles are the angles inside a triangle and that these angles add up to \begin{align*}180^\circ\end{align*}. This concept will now be extended to any polygon. As you can see in the images below, a polygon has the same number of interior angles as it does sides. But, what do the angles add up to?

Investigation 6-1: Polygon Sum Formula

Tools Needed: paper, pencil, ruler, colored pencils (optional)

1. Draw a quadrilateral, pentagon, and hexagon.

2. Cut each polygon into triangles by drawing all the diagonals from one vertex. Count the number of triangles.

Make sure none of the triangles overlap.

3. Make a table with the information below.

Name of Polygon Number of Sides Number of \begin{align*}\triangle s\end{align*} from one vertex (Column 3) \begin{align*}\times (^\circ\end{align*} in a \begin{align*}\triangle\end{align*}) Total Number of Degrees
Quadrilateral 4 2 \begin{align*}2 \times 180^\circ\end{align*} \begin{align*}360^\circ\end{align*}
Pentagon 5 3 \begin{align*}3 \times 180^\circ\end{align*} \begin{align*}540^\circ\end{align*}
Hexagon 6 4 \begin{align*}4 \times 180^\circ\end{align*} \begin{align*}720^\circ\end{align*}

4. Notice that the total number of degrees goes up by \begin{align*}180^\circ\end{align*}. So, if the number sides is \begin{align*}n\end{align*}, then the number of triangles from one vertex is \begin{align*}n - 2\end{align*}. Therefore, the formula would be \begin{align*}(n - 2) \times 180^\circ\end{align*}.

Polygon Sum Formula: For any \begin{align*}n-\end{align*}gon, the interior angles add up to \begin{align*}(n - 2) \times 180^\circ\end{align*}.

\begin{align*}\rightarrow n = 8& \\ (8 - 2) & \times 180^\circ\\ 6 & \times 180^\circ\\ 1& 080^\circ\end{align*}

Example 1: The interior angles of a polygon add up to \begin{align*}1980^\circ\end{align*}. How many sides does it have?

Solution: Use the Polygon Sum Formula and solve for \begin{align*}n\end{align*}.

\begin{align*}(n - 2) \times 180^\circ & = 1980^\circ\\ 180^\circ n - 360^\circ & = 1980^\circ\\ 180^\circ n & = 2340^\circ\\ n & = 13\end{align*}

The polygon has 13 sides.

Example 2: How many degrees does each angle in an equiangular nonagon have?

Solution: First we need to find the sum of the interior angles, set \begin{align*}n = 9.\end{align*}

\begin{align*}(9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ\end{align*}

“Equiangular” tells us every angle is equal. So, each angle is \begin{align*}\frac{1260^\circ}{9} = 140^\circ\end{align*}.

Equiangular Polygon Formula: For any equiangular \begin{align*}n-\end{align*}gon, the measure of each angle is \begin{align*}\frac{(n-2) \times 180^\circ}{n}\end{align*}.

If \begin{align*}m \angle 1 = m \angle 2 = m \angle 3 = m \angle 4 = m \angle 5 = m \angle 6 = m \angle7 = m \angle 8\end{align*}, then each angle is \begin{align*}\frac{(8 - 2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ\end{align*}

In the Equiangular Polygon Formula, the word equiangular can be switched with regular.

Regular Polygon: When a polygon is equilateral and equiangular.

Example 3: An interior angle in a regular polygon is \begin{align*}135^\circ\end{align*}. How many sides does this polygon have?

Solution: Here, we will set the Equiangular Polygon Formula equal to \begin{align*}135^\circ\end{align*} and solve for \begin{align*}n\end{align*}.

\begin{align*}\frac{(n - 2) \times 180^\circ}{n} & = 135^\circ\\ 180^\circ n - 360^\circ & = 135^\circ n\\ -360^\circ & = -45^\circ n\\ n & = 8 \qquad \quad \text{The polygon is an octagon}.\end{align*}

Example 4: Algebra Connection Find the measure of \begin{align*}x\end{align*}.

Solution: From our investigation, we found that a quadrilateral has \begin{align*}360^\circ\end{align*}.

Write an equation and solve for \begin{align*}x\end{align*}.

\begin{align*}89^\circ + (5x - 8)^\circ + (3x + 4)^\circ + 51^\circ & = 360^\circ\\ 8x & = 224^\circ\\ x & = 28^\circ\end{align*}

## Exterior Angles in Convex Polygons

An exterior angle is an angle that is formed by extending a side of the polygon (Chapter 4).

As you can see, there are two sets of exterior angles for any vertex on a polygon, one going around clockwise (\begin{align*}1^{st}\end{align*} hexagon), and the other going around counter-clockwise (\begin{align*}2^{nd}\end{align*} hexagon). The angles with the same colors are vertical and congruent.

The Exterior Angle Sum Theorem said the exterior angles of a triangle add up to \begin{align*}360^\circ\end{align*}. Let’s extend this theorem to all polygons.

Investigation 6-2: Exterior Angle Tear-Up

Tools Needed: pencil, paper, colored pencils, scissors

1. Draw a hexagon like the ones above. Color in the exterior angles.
2. Cut out each exterior angle.
3. Fit the six angles together by putting their vertices together. What happens?

The angles all fit around a point, meaning that the angles add up to \begin{align*}360^\circ\end{align*}, just like a triangle.

Exterior Angle Sum Theorem: The sum of the exterior angles of any polygon is \begin{align*}360^\circ\end{align*}.

Example 5: What is \begin{align*}y\end{align*}?

Solution: \begin{align*}y\end{align*} is an exterior angle and all the given angles add up to \begin{align*}360^\circ\end{align*}. Set up an equation.

\begin{align*}70^\circ + 60^\circ + 65^\circ + 40^\circ + y & = 360^\circ\\ y & = 125^\circ\end{align*}

Example 6: What is the measure of each exterior angle of a regular heptagon?

Solution: Because the polygon is regular, the interior angles are equal. It also means the exterior angles are equal. \begin{align*}\frac{360^\circ}{7} \approx 51.43^\circ\end{align*}

Know What? Revisited The stars make a pentagon and an octagon. The pentagon looks to be regular, but we cannot tell without angle measurements or lengths.

## Review Questions

• Questions 1-13 are similar to Examples 1-3 and 6.
• Questions 14-30 are similar to Examples 4 and 5.
1. Fill in the table.
# of sides # of \begin{align*}\triangle s\end{align*} from one vertex \begin{align*}\triangle s \times 180^\circ\end{align*} (sum) Each angle in a regular \begin{align*}n-\end{align*}gon Sum of the exterior angles
3 1 \begin{align*}180^\circ\end{align*} \begin{align*}60^\circ\end{align*}
4 2 \begin{align*}360^\circ\end{align*} \begin{align*}90^\circ\end{align*}
5 3 \begin{align*}540^\circ\end{align*} \begin{align*}108^\circ\end{align*}
6 4 \begin{align*}720^\circ\end{align*} \begin{align*}120^\circ\end{align*}
7
8
9
10
11
12
1. Writing Do you think the interior angles of a regular \begin{align*}n-\end{align*}gon could ever be \begin{align*}180^\circ\end{align*}? Why or why not? What about \begin{align*}179^\circ\end{align*}?
2. What is the sum of the angles in a 15-gon?
3. What is the sum of the angles in a 23-gon?
4. The sum of the interior angles of a polygon is \begin{align*}4320^\circ\end{align*}. How many sides does the polygon have?
5. The sum of the interior angles of a polygon is \begin{align*}3240^\circ\end{align*}. How many sides does the polygon have?
6. What is the measure of each angle in a regular 16-gon?
7. What is the measure of each angle in an equiangular 24-gon?
8. Each interior angle in a regular polygon is \begin{align*}156^\circ\end{align*}. How many sides does it have?
9. Each interior angle in an equiangular polygon is \begin{align*}90^\circ\end{align*}. How many sides does it have?
10. What is the measure of each exterior angle of a dodecagon?
11. What is the measure of each exterior angle of a 36-gon?
12. What is the sum of the exterior angles of a 27-gon?

Algebra Connection For questions 14-26, find the measure of the missing variable(s).

1. The interior angles of a pentagon are \begin{align*}x^\circ, x^\circ, 2x^\circ, 2x^\circ,\end{align*} and \begin{align*}2x^\circ\end{align*}. What is \begin{align*}x\end{align*}?
2. The exterior angles of a quadrilateral are \begin{align*}x^\circ, 2x^\circ, 3x^\circ,\end{align*} and \begin{align*}4x^\circ.\end{align*} What is \begin{align*}x\end{align*}?
3. The interior angles of a hexagon are \begin{align*}x^\circ, (x + 1)^\circ, (x + 2)^\circ, (x + 3)^\circ, (x + 4)^\circ,\end{align*} and \begin{align*}(x + 5)^\circ.\end{align*} What is \begin{align*}x\end{align*}?

1. \begin{align*}180^\circ\end{align*}
2. \begin{align*}72^\circ + (7x+3)^\circ + (3x+5)^\circ = 180^\circ\!\\ {\;}\qquad \qquad \qquad \quad \ 10x + 80^\circ = 180^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \ \ \quad \ 10x = 100^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \ \ \qquad \ x = 10^\circ\end{align*}
3. \begin{align*}180^\circ\end{align*}
1. \begin{align*}w = 108^\circ, \ x = 49^\circ, \ y = 131^\circ, \ z = 121^\circ\end{align*}
2. \begin{align*}360^\circ\end{align*}

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