Polygons can be CONCAVE or CONVEX CONCAVE CONVEX

Concave and Convex Polygons If a polygon has an indentation (or cave), the polygon is called a concave polygon. Any polygon that does not have an indentation is called a convex polygon. Any two points in the interior of a convex polygon can be connected by a line segment that does not cut or cross a side of the polygon. Concave polygon Convex polygon We will only be discussing CONCAVE polygons

Triangle Octagon Quadrilateral Nonagon Pentagon Decagon Hexagon Dodecagon Heptagon Hendecagon n-gon

Important Terms A VERTEX is the point of intersection of two sides F A segment whose endpoints are two nonconsecutive vertices is called a DIAGONAL. A B CONSECUTIVE VERTICES are two endpoints of any side. C E D Sides that share a vertex are called CONSECUTIVE SIDES.

Tear off two vertices….

Line up the 3 angles (all vertices touching)

A straight line = 180°

Angle sum of a Triangle n n 180 <1 + <2 + <3 = 180 2 ALWAYS!!! 1 3

Consider a Quadrilateral n What is the angle sum? <1 + <2 + <3 + <4 = ?

Quadrilateral n Draw a diagonal…what do you get? 2 3 5 1 4 Two triangles 6

Quadrilateral n Each triangle = 180 2 3 180 1 4 Therefore the two triangles together = 360 5 180 6

Angle sum of a Quadrilateral n 180 + 180 = 360

Consider a Pentagon n What is the angle sum?

Pentagon n Draw the diagonals from 1 vertex How many triangles?

Angle sum of a Pentagon n Draw the diagonals from 1 vertex 180 180

Continue this process through Decagon n Draw the diagonals from 1 vertex

What about a 52 -gon? n n What is the angle sum? Can you find the pattern? 1 180 2 360 3 540 4 720 5 900 6 1080

Find the nth term 7 1260 8 1440 n-2 (n – 2)(180)

pentagon Find m 1. (4 x + 15) 2 (5 x - 5) 1 (8 x - 10) m 1 = 5(20) - 5 = 95 3 540 17 x + 200= 540 110 -200 17 x = 340 17 17 5 x - 5 + 4 x + 15 + 8 x - 10 + 110 + 90 = x = 20

More important terms Interior Angles Exterior Angles the SUM of an interior angle and it’s corresponding exterior angle = 180 o

Sums of Exterior Angles 180 4 2 3 1 180 5 6 180 • 3 = 540 Sum of Interior & Exterior Angles =540 Sum of Interior Angles =180 Sum of Exterior Angles = 540 - 180 =360

Sums of Exterior Angles 180 180 • 4 = 720 Sum of Interior & Exterior Angles =720 Sum of Interior Angles =360 Sum of Exterior Angles = 720 - 360 =360

Sums of Exterior Angles Sum of Interior & Exterior Angles = Sum of Interior Angles = Sum of Exterior Angles = 180 • 5 = 900 180 • 3 = 540 900 - 540 =360

What conclusion can you come up with regarding the exterior angle sum of a CONVEX npolygon? ? Sum of Interior & Exterior Angles = 180 n Sum of Interior Angles = 180(n-2) = 180 n - 360 Sum of Exterior Angles = 180 n – (180 n – 360)

The exterior angle sum of a CONVEX polygon = 360

Important Terms EQUILATERAL - All sides are congruent EQUIANGULAR - All angles are congruent REGULAR - All sides and angles are congruent

Interior Angle Measure of a REGULAR polygons 60 Equilateral Triangle Angle measure = 60 90 Square Angle measure = 90 These are measurement that we generally know at this time, But what about the other regular polygons? How do we calculate the interior angle measure?

Pentagon 72 108 72 108 72

Interior Angle Measure of a REGULAR polygons 108 120 Calculate by: Angle Sum Number of sides 135