To Classify polygons To find the sums of the measures of the interior

and exterior angles of polygons.

Polygon—a plane figure that meets the following conditions: It is formed by 3 or more segments called sides, such that no

two sides with a common endpoint are collinear. Each side intersects exactly two other sides, one at each

endpoint.

Vertex – each endpoint of a side. Plural is vertices. You can name a polygon by listing its vertices consecutively. For instance, PQRST and QPTSR are two correct names for the polygon above.

VERTEX

VERTEX

Q

R

ST

PSIDE

State whether the figure is a polygon. If it is not, explain why.

Not D – has a side that isn’t a segment – it’s an arc.

Not E– because two of the sides intersect only one other side.

Not F because some of its sides intersect more than two sides/

F

E

D

CBA

Figures A, B, and C are polygons.

Number of sides Type of Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

Number of sides Type of Polygon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n-gon

Convex if no line that contains a side of the polygon contains a point in the interior of the polygon.

Concave or non-convex if a line does contain a side of the polygon containing a point on the interior of the polygon.

See how it doesn’t go on theInside-- convex

See how this crossesa point on the inside?Concave.

Identify the polygon and state whether it is convex or concave.

A polygon is EQUILATERALIf all of its sides are congruent.A polygon is EQUIANGULARif all of its interior angles are congruent. A polygon is REGULAR if it isequilateral and equiangular.

x°+ 2x° + 70° + 80° = 360°

3x + 150 = 360

3x = 210

x = 70

Sum of the measures of int. s of a quadrilateral is 360°

Combine like terms

Subtract 150 from each side.

Divide each side by 3.

P

Q

S

R

80°

70°

2x°x°

Find m Q and mR.

mQ = x° = 70°

mR = 2x°= 140°

►So, mQ = 70° and mR = 140°

Sketch polygons with 4, 5, 6, 7, and 8 sides

Divide Each Polygon into triangles by drawing all diagonals that are possible from one vertex

Multiply the number of triangles by 180 to find the sum of the measures of the angles of each polygon.

1) Look for a pattern. Describe any that you have found.

2) Write a rule for the sum of the measures of the angles of an n-gon

The sum of the measures of the angles of an n-gon is

(n-2)180

Ex: Find the sum of the measures of the angles of a 15-gon

Sum = (n-2)180 = (15-2)180 = 13*180

= 2340

The sum of the interior angles of a polygon is 9180. How many sides does the polygon have?

Sum = (n-2)180 9180 = (n-2)180 51 = n-2 53 = n The polygon has 53 sides.

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

An equilateral polygon has all sides congruent

An equiangular polygon has all angles congruent

A regular polygon is both equilateral and equiangular.

The measure of an exterior angle of a regular polygon is 36. Find the measure of an interior angle, and find the number of sides.

Exterior angles = 360 Since regular, n*36 = 360 n = 10 Since exterior angle = 36,

interior angle 180-26 = 144