6.1: Angles in Polygons
Learning Objectives
 Extend the concept of interior and exterior angles from triangles to convex polygons.
 Find the sums of interior angles in convex polygons.
 Identify the special properties of interior angles in convex quadrilaterals.
Review Queue
 Find the measure of and .

 Find , and .
 What is ?
 What two angles add up to ?
 What are , and called? What are , and called?
Know What? To the right is a picture of Devil’s Post pile, near Mammoth Lakes, California. These posts are cooled lava (called columnar basalt) and as the lava pools and cools, it ideally would form regular hexagonal columns. However, variations in cooling caused some columns to either not be perfect or pentagonal.
First, define regular in your own words. Then, what is the sum of the angles in a regular hexagon? What would each angle be?
Interior Angles in Convex Polygons
Recall from Chapter 4, that interior angles are the angles inside a closed figure with straight sides. Even though this concept was introduced with triangles, it can 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.
From Chapter 1, we learned that a diagonal connects two nonadjacent vertices of a convex polygon. Also, recall that the sum of the angles in a triangle is . What about other polygons?
Investigation 61: 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 from one vertex  (Column 3) ( in a )  Total Number of Degrees 

Quadrilateral  4  2  
Pentagon  5  3  
Hexagon  6  4 
4. Do you see a pattern? Notice that the total number of degrees goes up by . So, if the number sides is , then the number of triangles from one vertex is . Therefore, the formula would be .
Polygon Sum Formula: For any gon, the sum of the interior angles is .
Example 1: Find the sum of the interior angles of an octagon.
Solution: Use the Polygon Sum Formula and set .
Example 2: The sum of the interior angles of a polygon is . How many sides does this polygon have?
Solution: Use the Polygon Sum Formula and solve for .
Example 3: How many degrees does each angle in an equiangular nonagon have?
Solution: First we need to find the sum of the interior angles in a nonagon, set .
Second, because the nonagon is equiangular, every angle is equal. Dividing by 9 we get each angle is .
Equiangular Polygon Formula: For any equiangular gon, the measure of each angle is .
Regular Polygon: When a polygon is equilateral and equiangular.
It is important to note that in the Equiangular Polygon Formula, the word equiangular can be substituted with regular.
Example 4: Algebra Connection Find the measure of .
Solution: From our investigation, we found that a quadrilateral has . We can write an equation to solve for .
Exterior Angles in Convex Polygons
Recall that an exterior angle is an angle on the outside of a polygon and 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. It does not matter which set you use because one set is just the vertical angles of the other, making the measurement equal. In the picture to the left, the colormatched angles are vertical angles and congruent.
In Chapter 4, we introduced the Exterior Angle Sum Theorem, which stated that the exterior angles of a triangle add up to . Let’s extend this theorem to all polygons.
Investigation 62: Exterior Angle TearUp
Tools Needed: pencil, paper, colored pencils, scissors
 Draw a hexagon like the hexagons above. Color in the exterior angles as well.
 Cut out each exterior angle and label them 16.
 Fit the six angles together by putting their vertices together. What happens?
The angles all fit around a point, meaning that the exterior angles of a hexagon add up to , just like a triangle. We can say this is true for all polygons.
Exterior Angle Sum Theorem: The sum of the exterior angles of any polygon is .
Proof of the Exterior Angle Sum Theorem
Given: Any gon with sides, interior angles and exterior angles.
Prove: exterior angles add up to
NOTE: The interior angles are .
The exterior angles are .
Statement  Reason 

1. Any gon with sides, interior angles and exterior angles.  Given 
2. and are a linear pair  Definition of a linear pair 
3. and are supplementary  Linear Pair Postulate 
4.  Definition of supplementary angles 
5.  Sum of all interior and exterior angles in an gon 
6.  Polygon Sum Formula 
7.  Substitution PoE 
8.  Distributive PoE 
9.  Subtraction PoE 
Example 5: What is ?
Solution: is an exterior angle, as well as all the other given angle measures. Exterior angles add up to , so set up an equation.
Example 6: What is the measure of each exterior angle of a regular heptagon?
Solution: Because the polygon is regular, each interior angle is equal. This also means that all the exterior angles are equal. The exterior angles add up to , so each angle is .
Know What? Revisited A regular polygon has congruent sides and angles. A regular hexagon has total degrees. Each angle would be divided by 6 or .
Review Questions
 Fill in the table.
# of sides  # of from one vertex  (sum)  Each angle in a regular gon  Sum of the exterior angles 

3  1  
4  2  
5  3  
6  4  
7  
8  
9  
10  
11  
12 
 What is the sum of the angles in a 15gon?
 What is the sum of the angles in a 23gon?
 The sum of the interior angles of a polygon is . How many sides does the polygon have?
 The sum of the interior angles of a polygon is . How many sides does the polygon have?
 What is the measure of each angle in a regular 16gon?
 What is the measure of each angle in an equiangular 24gon?
 What is the measure of each exterior angle of a dodecagon?
 What is the measure of each exterior angle of a 36gon?
 What is the sum of the exterior angles of a 27gon?
 If the measure of one interior angle of a regular polygon is , how many sides does it have?
 How many sides does a regular polygon have if the measure of one of its interior angles is ?
 If the measure of one interior angle of a regular polygon is , how many sides does it have?
 How many sides does a regular polygon have if the measure of one exterior angle is ?
 If the measure of one exterior angle of a regular polygon is , how many sides does it have?
 How many sides does a regular polygon have if the measure of one exterior angle is ?
For questions 1120, find the measure of the missing variable(s).
 The interior angles of a pentagon are , and . What is the measure of the larger angles?
 The exterior angles of a quadrilateral are , and . What is the measure of the smallest angle?
 The interior angles of a hexagon are , and . What is ?
 Challenge Each interior angle forms a linear pair with an exterior angle. In a regular polygon you can use two different formulas to find the measure of each exterior angle. One way is and the other is ( minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.
 Angle Puzzle Find the measures of the lettered angles below given that .
Review Queue Answers
 interior angles, exterior angles