Geometry

3.5 Angles of a PolygonStandard 12.0 & 13.0

Polygons (“many angles”)

• have vertices, sides, angles, and exterior angles

• are named by listing consecutive vertices in order A B

C

DE

F

Hexagon ABCDEF

Polygons

• formed by line segments, no curves• the segments enclose space

• each segment intersects two other segments

Not PolygonsPolygons

Diagonal of a PolygonA segment connecting two nonconsecutive vertices

Diagonals

Convex Polygons

No side ”collapses” in toward the center

Easy test : RUBBER BAND stretched around the figure would have the same shape…….

NonconvexPolygons

Convex Polygons

When the textbook refers to polygons, it means convex polygons

From now on…….

Polygons are classified by number of sides

Number of sides Name of Polygon

3 triangle4 quadrilateral5 pentagon6 hexagon8 octagon10 decagonn n-gon

Interior Angles of a Polygon•To find the sum of angle measures, divide the polygon into triangles•Draw diagonals from just one vertex

4 sides, 2 trianglesAngle sum = 2 (180)

5 sides, 3 trianglesAngle sum = 3 (180)

6 sides, 4 trianglesAngle sum = 4 (180)

DO YOU SEE A PATTERN ?

Interior Angles of a Polygon

4 sides, 2 trianglesAngle sum = 2 (180)

5 sides, 3 trianglesAngle sum = 3 (180)

6 sides, 4 trianglesAngle sum = 4 (180)

The pattern is:

ANGLE SUM = (Number of sides – 2) (180)

Theorem

The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180.

Example: 5 sides. 3 triangles.Sum of angle measures is

(5-2)(180) = 3(180) = 540

Exterior Angles of a Polygon

1

2

3

4

5

Draw the exterior angles

12

3

45

Put them togetherThe sum = 360Works with every polygon!

Theorem

The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.

If a polygon is both equilateral and equiangular it is called a regular polygon

Regular Polygons

120

120

120

120

120

120

Equilateral Equiangular

120

120

120

120

120

120

Regular

Example 1

A polygon has 8 sides (octagon.) Find:

a) The interior angle sum

b) The exterior angle sum

n=8, so (8-2)180 = 6(180) = 1080

360

Example 2

Find the measure of each interior and exterior angle of a regular pentagon

Interior: (5-2)180 = 3(180) = 540540 = 108 each 5

Exterior: 360 = 72 each 5

Example 3

How many sides does a regular polygon have if:

a) the measure of each exterior angle is 45

360 = 45 360 = 45n n n = 8 8 sides: an

octagon

b) the measure of each interior angle is 150

(n-2)180 = 150 (n-2)180 = 150n n 180n – 360 = 150n

- 360 = - 30n n = 12 12

sides

Homework