polygons can be concave or convex convex concave

Polygons can be CONCAVE or CONVEX

CONVEX

CONCAVE

Concave and ConvexPolygons

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

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Dodecagon

n-gonHendecagon

F

A B

C

DE

Important TermsA VERTEX is the point of intersection of two sides

A segment whose

endpoints are two

nonconsecutive vertices is

called a DIAGONAL.

CONSECUTIVE VERTICES are two endpoints of any side.

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 180° <1 + <2 + <3 = 180°

1

2

3

ALWAYS!!!

Consider a Quadrilateral What is the angle sum?

<1 + <2 + <3 + <4 = ?

Quadrilateral Draw a diagonal…what do you

get?

Two triangles

1

2 3

4

5

6

Quadrilateral Each triangle = 180°

Therefore the two triangles together = 360°

1

2 3

4

5

6

180°

180°

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 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)

m1 =

1

2

3

110

(5x - 5)

(4x + 15)

(8x - 10)

pentagon

5x - 5 + 4x + 15 + 8x - 10 + 110 + 90 =

54017x + 200= 540 -200 -200

17x = 340

x = 20 17 17

5(20) - 5

= 95

Find m1.

More important terms

Exterior Angles

Interior Angles

the SUM of an interior angle and it’s corresponding exterior angle = 180o

Sum of Interior Angles =

Sum of Interior & Exterior Angles =

180

12

34

5 6

180

180

180

540

Sum of Exterior Angles =

360 540- 180=

Sums of Exterior Angles

180•3 = 540

180

180

180

180

Sum of Interior Angles =

Sum of Interior & Exterior Angles =

360 720

Sum of Exterior Angles =

360 720- 360=

Sums of Exterior Angles

180•4 = 720

Sums of Exterior Angles

Sum of Interior & Exterior Angles =Sum of Interior Angles =

Sum of Exterior Angles =

180•5 =

180•3 =

900

540

900- 540=360

What conclusion can you come up with regarding the exterior angle sum of a CONVEX n-polygon??

Sum of Interior & Exterior Angles =Sum of Interior Angles =

Sum of Exterior Angles =

180n

180(n-2) = 180n - 360

180n – (180n – 360)

The exterior angle sum of a CONVEX polygon =

360°

Important Terms

EQUILATERAL - All sides are congruentEQUIANGULAR - All angles are congruentREGULAR - 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

108°

108°

108°

108°

108° 72°

72°

72°

72°

72°

Interior Angle Measure of a REGULAR polygons

108°

120°

135°

Calculate by:

Angle Sum

Number of sides