1. 6-1 Angles of Polygons The student will be able to:
1. Find the sum of the measures of the interior angles of a polygon.
2. Find the sum of the measures of the exterior angles of a polygon.

2. A Polygon is a closed figure with sides constructed from segments
A Polygon is a closed figure with sides constructed from segments. Polygons can be convex or concave and regular or irregular. Tell whether the following figures are polygons. If so, tell whether each is convex or concave and regular or irregular.
yes
yes
yes
convex
convex
convex
regular
regular
irregular
yes no
concave
irregular

3. Interior Angle sum Theorem – If a convex polygon has n sides and S is the sum of the measures of its interior angles, then S = 180(n – 2).
Convex Polygon
Number of Sides (n)
Sum of Angles (S)
Triangle 3
180(3 – 2) = 180
Quadrilateral 4
180(4 – 2) = 360
Pentagon 5
180(5 – 2) = 540
Hexagon 6
180(6 – 2) = 720
Heptagon 7
180(7 – 2) = 900
Octagon 8
180(8 – 2) = 1080
n sides n
180(n – 2) = 180n - 360

4. How many sides do we have? 4
Example 1: Find the measure of each interior angle of quadrilateral ABCD.
How many sides do we have? 4
How many degrees are in the quadrilateral?
360
How do we find the sum of the interior angles?
360 = x + x
= 135
360 = x
-180
180 = 4x
= 45
45 = x

5. How many sides do we have? 5
Find the measure of each interior angle of pentagon HJKLM shown = 10x
How many sides do we have? 5
How many degrees are in a pentagon?
540
How do we find the sum of the interior angles?
540 = x + 2x + (3x + 14) + (3x + 14)
540 =
170 +
10x
= 74
-170
370 = 10x
= 74
37 = x
= 125
= 125

6. sum of interior angle measures = each angle measure
Example 2: The measure of an interior angle of a regular polygon is Find the number of sides in the polygon.
sum of interior angle measures
= each angle measure
number of congruent angles
180(n – 2)
= 144 n
180n – 360 = 144n
-180n
-360 = -36n
10 = n

7. 360 _ number of congruent angles = each angle measure
The sum of the measures of Exterior Angles Exterior Angle Sum Theorem – If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360°.
No matter how many sides there are in the convex polygon, the sum of the exterior angles of that polygon equals 360°.
_ number of congruent angles
= each angle measure
The sum of an interior angle and an exterior angle is 180°.

8. Example 4: Find the value of x in the diagram.
x + 9x + 2x
= 360
x = 360
-139
17x = 221
x = 13
Example 5: Find the measure of each exterior angle of a regular dodecagon.
How many sides are in a dodecagon? 12
How many degrees are in the exterior angles of a dodecagon?
360
How do we find the measure of each exterior angle?
360_______ # of congruent angles
= the measure of each angle