sum of interior and exterior angles in polygons

polygons

Submitted By Aneesha mol k.pb.Ed mathematicsCandidate code :18015352002

Essential Question – How can I find angle measures in polygons without using a protractor?

PolygonsA polygon is a closed figure formed by a finite

number of segments such that:1. the sides that have a common endpoint

are noncollinear, and2. each side intersects exactly two other

sides, but only at their endpoints.

Nonexamples

PolygonsCan be concave or convex.

Concave Convex

Polygons are named by number of sidesNumber of Sides Polygon

34

567891012n

TriangleQuadrilateralPentagonHexagonHeptagonOctagonNonagonDecagonDodecagon

n-gon

Regular PolygonA convex polygon in which all the sides are

congruent and all the angles are congruent is called a regular polygon.

Draw a: Quadrilateral Pentagon Hexagon Heptagon Octogon

Then draw diagonals to create triangles.A diagonal is a segment connecting two

nonadjacent vertices (don’t let segments cross)

Add up the angles in all of the triangles in the figure to determine the sum of the angles in the polygon.

Complete this tablePolygon # of

sides# of triangles Sum of

interior angles

Polygon # of sides

# of triangles

Sum of interior angles

TriangleQuadrilater

alPentagonHexagonHeptagonOctagon

n-gon

345

678n

34

56

n - 2

21 180°

2 · 180 = 360°3 · 180 = 540°4 · 180 = 720°

5 · 180 = 900°

6 · 180 = 1080°(n – 2) · 180°

Polygon Interior Angles TheoremThe sum of the measures of the interior

angles of a convex n-gon is (n – 2) • 180.Examples – 1.Find the sum of the measures of the interior

angles of a 16–gon.2.If the sum of the measures of the interior angles

of a convex polygon is 3600°, how many sides does the polygon have.

3.Solve for x.

4x - 2

82

108

2x + 10

(16 – 2)*180

(n – 2)*180 = 3600

180n – 360 = 3600 + 360 + 360

180n = 3960 180 180 n = 22 sides

(4 – 2)*180 = 360

108 + 82 + 4x – 2 + 2x + 10 = 3606x + 198 = 360

6x = 162 6 6

x = 27

= 2520°

Draw a quadrilateral and extend the sides.There are two sets of angles formed when the sides of a polygon are extended. • The original angles are called interior angles. • The angles that are adjacent to the interior angles are called exterior angles.These exterior angles can be formed when any side is extended. What do you notice about the interior angle and the exterior angle?

What is the measure of a line?

What is the sum of an interior angle with the exterior angle?

They form a line.

180°

180°

If you started at Point A, and followed along the sides of the quadrilateral making the exterior turns that are marked, what would happen?You end up back where you started or you would make a circle.

What is the measure of the degrees in a circle?

A

BC

D

360°

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

Each exterior angle of a regular polygon is 360

n

where n is the number of sides in the polygon

Polygon Exterior Angles Theorem

54⁰

68⁰

65⁰

(3x + 13)⁰

60⁰

(4x – 12)⁰

Find the value for x. Sum of exterior angles is 360°

(4x – 12) + 60+ (3x + 13) + 65 + 54+ 68 = 360 7x + 248 = 360 – 248 – 248 7x = 112 7 7 x = 12

Example

What is the sum of the exterior angles in an octagon?

What is the measure of each exterior angle in a regular octagon?

360°

360°/8

= 45°

Classwork/HomeworkTextbook: Read and study p298-299

Complete p300-301 (1-21)

Show your work!