Angles of Polygons

Polygons can be CONCAVE or 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 Dodecagon Hexagon n-gon Heptagon Hendecagon

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

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

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

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

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

Angle sum of a Quadrilateral 180° + 180° = 360°

Consider a Pentagon What is the angle sum?

Pentagon Draw the diagonals from 1 vertex How many triangles?

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

Continue this process through Decagon Draw the diagonals from 1 vertex

Continue this process through Decagon Draw the diagonals from 1 vertex

What about a 52-gon? 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 5(20) - 5 m1 = = 95 Find m1. (4x + 15) 2 (5x - 5) 3 110 (5x - 5) (4x + 15) (8x - 10) m1 = 5(20) - 5 = 95 540 17x + 200= 540 -200 -200 17x = 340 5x - 5 + 4x + 15 + 8x - 10 + 110 + 90 = 17 17 x = 20

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

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

Sums of Exterior Angles 180 180 180 180•4 = 720 180 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 = 180•5 = 900 Sum of Interior Angles = 180•3 = 540 Sum of Exterior Angles = 360 900- 540=

What conclusion can you come up with regarding the exterior angle sum of a CONVEX n-polygon?? Sum of Interior & Exterior Angles = 180n Sum of Interior Angles = 180(n-2) = 180n - 360 Sum of Exterior Angles = 180n – (180n – 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° 90° Equilateral Triangle Angle measure = 60° 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° 108° 72° 72° 72° 108° 108°

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