1. 6.1 Notes: Angles of Polygons
EQ: What is the sum of the measures of the interior angles of a polygon? Of the exterior angles of a polygon?

2. Quadrilaterals
Parallelogram
Square
Trapezoid
Rectangle
Rhombus
Kite

3. Vocab! Diagonal of a Polygon
A segment that connects any two nonconsecutive vertices

4. Sum Sum Sum of the Angles of a Polygon
The _______ of the angle measures of a polygon is the __________ of the angle measures of the triangles formed by drawing all the possible diagonals from one vertex.
Polygon Interior Angles Theorem
The sum of the interior angle measures of an n-sided convex polygon is (n-2) ∘ 180
Sum
Sum
Polygon
Number of Sides
Number of Triangles
Sum of Interior Angle Measures
Triangle 3 1
180 or 180
Quadrilateral 4 2
180 or 360
Pentagon 5
180 or 540
Hexagon 6
180 or 720
n – gon n
n-2
(n – 2)180

5. Example 1
Find the sum of the measures of the interior angles of a convex nonagon.
Nonagon = 9 sides
Equation = (n – 2) ∘ 180
n = 9
(9 – 2) ∘ 180
Sum = 1260

6. Example 2
Find the sum of the measures of the interior angles of a convex 11-gon.
11-gon = 11 sides
Equation = (n – 2) ∘ 180
n = 11
(11 – 2) ∘ 180
Sum = 1,620

7. Example 3 Find the value of x in the diagram. Equation = (n – 2) ∘ 180
(4 – 2) ∘ 180
Sum = 360
x = 360
288 + x = 360
x = 72

8. Do the you try on your own first before you look at the answers!

9. You Try!
Find the measures of each interior angle of parallelogram RSTU.
Parallelogram = 4 sides which is 360°
11x x x + 5x = 360
32x + 8 = 360
x = 11

10. Example 4
a) The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon. b) The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.
All angles of a regular polygon are equal.
Sum of the angles = 150n
150n = (n-2) ∘ 180
150n = 180n – 360
-30n = -360
n = 12
Sum of the angles = 144n
144n = (n-2) ∘ 180
144n = 180n – 360
-36n = -360
n = 10

11. You Try!
1. The sum of the measures of the interior angles of a convex polygon is 900˚. Classify the polygon by the number of sides.
Sum of the angles = 900
900 = (n-2) ∘ 180
900 = 180n – 360
1260 = 180n
n = 7

12. Using the polygons below, does a relationship exist between the number of sides and sum of its exterior angles?

13. Polygon Exterior Angles Theorem
The sum of the exterior angle measures of a convex polygon, one angle at each vertex, is 360.
Examples of finding exterior angles
Example 5: Find x in the diagram
Example 6: Find the measure of each exterior angle of a regular decagon.
Add up all of the exterior angles to get one equation.
31x – 12 = 360
X=12
Decagon = 10 sides
360/10 36

14. You Try! 1. Find the value of x in the diagram. 2x + x + 89 + 67 = 360