Chapter 3

Polygons

I. Properties of a Polygon

A. A plane figure formed by three or more consecutive segments (sides or laterals)

B. Each side intersects exactly two other sides at its endpoints.

C. These intersections are called vertices.

D. No three consecutive vertices are collinear.

3.1 Basic Polygons

II. Types of Polygons3 sides – triangle 4 sides – quadrilateral5 sides – pentagon6 sides – hexagon 7 sides – heptagon8 sides – octagon9 sides – nonagon10 sides – decagon12 sides – dodecagon20 sides – icosagon n sides – n-gon

III. Convex and Concave Polygons

A. Convex – any two points inside of the polygon can be connected with a segment that is completely inside the polygon.

B. Concave – opposite of convex

IV. Regular Polygons

A. Polygons that are both equilateral and equiangular.

B. Equilateral – all the sides are the same length

C. Equiangular – all the angles are the same measure.

D. Perimeter – the distance around a polygon

Regular Pentagon

Diagonal – a segment connecting two non-consecutive vertices

3.2 Angles of Polygons

I. Polygon Interior Angle Theorem

5 sides -- pentagon

3 triangles x 180

540°

3.2 Angles of Polygons

I. Polygon Interior Angle Theorem

6 sides -- hexagon

4 triangles x 180

720°

Total number of degrees in a polygon = 180(n – 2)

3.2 Angles of Polygons

I. Polygon Interior Angle Theorem

6 sides -- hexagon

4 triangles x 180

720°

110°

122°

95°

103°x

x + 30

II. Regular Polygons

180 (3)

540° 5

108°

Each angle of a regular polygon = 180(n – 2) n

180 (n – 2)

Regular Octagon

180 ( n – 2)

180 ( 6)

1080° 8

135°

III. Polygon Exterior Angle Theorem

108°

108°

108°108°

108°

72°

72°

72°

72°

72°

The sum of the measures of the exterior angles of a polygon is 360°

3.3 Types of Quadrilaterals

I. Parallelogram - quadrilateral with opposite sides that are parallel and congruent

125°

125°55°

55°

Opposite angles are equal in measure

II. Rectangle

A parallelogram with four right angles.

III. Rhombus

Parallelogram with four congruent sides

120°

120°

60°

60°

IV. Square

Parallelogram with four congruent sides and four right angles

A square is a parallelogram, a rectangle, and a rhombus.

V. Trapezoid

A quadrilateral with one pair of parallel sides

3.4 Trapezoids

Parallel sides are called the bases.

Non-parallel sides are called the legs.

I. Special Trapezoids

A. Isosceles Trapezoid

a trapezoid with two congruent sides

70° 70°

110° 110°

Base angles are congruent

I. Special Trapezoids

B. Right Trapezoid

a trapezoid with two right angles

125°

55°

II. Trapezoid Midsegment Theorem

A B

C D

E F

25 cm.

39 cm.

32 cm.

MS = (sum of bases)2

MS = (25 + 39)2

MS = (64)2MS = 32