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?

Quadrilaterals Parallelogram Square Trapezoid Rectangle Rhombus Kite

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

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

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

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

Example 3 Find the value of x in the diagram. Equation = (n – 2) ∘ 180 (4 – 2) ∘ 180 Sum = 360 108 + 121 + 59 + x = 360 288 + x = 360 x = 72

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

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

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

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

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

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

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