Sum of Interior Angles of a Polygon

•Th. 6.1 – Polygon Interior Angles Theorem

•The sum of the measures of the interior angles of a convex n-gon is 180o(n – 2)

•Let = sum = 180o(n – 2)

Sum of Interior Angles of a Polygon

•Corollary to Th. 6.1

•The measure of each interior angle of a regular n – gon is 1/n(180o)(n – 2)

•m = 1/n(180o)(n – 2)

•Th. 6.2 Polygon Exterior Angles Theorem –

•The sum of the measures of the exterior angles, one from each vertex, of a convex polygon is 360o.

= 360

•Corollary to Th. 6.2 –

•The measure of each exterior angle of a regular n – gon is 1/n(360o).

•m = 1/n(360o).

Ex. 1. The measure of each angle of a regular n – gon is 160o. How many sides does the polygon have? (what is n?)

Ex. 2. The measure of each exterior angle of a regular polygon is 72o. How many sides does the polygon have?

Ex. 3 Find the measure of each interior angle of the quadrilateral shown below.

A B

CD

x + 30x + 90

x + 60 x

276/ 11 – 33 odd, 34

You are shown part of a convex n-gon. The pattern of congruent angles continues around the polygon. Find n. (hint, consider exterior angles)

142o

158o

Parallelograms

Proving Quadrilaterals are Parallelograms

W12

oN

– 5

62.5

8’

A B

CD

Definition – If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.

AB || DC, AD || BC

A B

CD

Th. 6.7 – If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

AB DC, AD BC

A B

CD

Th. 6.8 – If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

A C, B D

A B

CD

Th. 6.9 – If an angle of a quadrilateral is supplementary to both consecutive angles, then the quadrilateral is a parallelogram.

A is supplementary to B and D

A B

CD

Th. 6.10 – If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

AC bisects DB, DB bisects AC,

AE EC, BE ED

E

A B

CD

Th. 6.11 – If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram.

AB DC, AB || DC

P Q

U

RS

G:, PU UR P: PQRS is a parallelogram

1., PT UR 1. Given

If diagonals bisect each other the quadrilateral is a parallelogram

290/1 – 18, 20, 22 – 26

Sid Gilman

Bill Walsh

Sam Wyche Fassel

Mike Holmgren

Dennis Green

George Seifert

Wayne Coslet

MikeShanahan

BradMusgrave

Steve Mariucci

RayRhodes

MikeSherman

Quadrilaterals

Parallelograms TrapezoidsPerpendicular

DiagonalsOthers

No Properties

Rectangles Rhombus

Squares

Kites Others

TT

Special Parallelograms

Rhombus – a parallelogram that has all 4 sides congruent.

Rectangle – a parallelogram that has 4 right angles.

Square – a parallelogram that is both a rhombus and a rectangle.

Proving Special Parallelograms

6.12 – A parallelogram is a rhombus iff its diagonal are perpendicular.

6.13 – A parallelogram is a rhombus iff, each diagonal bisects a pair of opposite angles.

6.15 – A quadrilateral is a rhombus iff it has 4 congruent sides.

6.14 – A parallelogram is a rectangle iff, its diagonals are congruent.

CD

BA

AC BD

6.16 – A quadrilateral is a rectangle iff it has 4 right angles.

Proving Special Parallelograms

296/1 – 12, 21 – 24, 34

Trapezoids

A quadrilateral with exactly one pair of parallel sides.

Base

Base

LegLeg

Isosceles Trapezoid – Legs are congruent

Trapezoid has two pairs of base angles

Trapezoids

6.17 – If a trapezoid is isosceles, then each pair of base angles is congruent.

6.19 – If a trapezoid has one pair of congruent base angles, then it is isosceles.

6.18 - If a trapezoid is isosceles, then its diagonals are congruent.

6.20 – If a trapezoid has congruent diagonals, then it is isosceles.

Trapezoids

YX

D C

BA

Midsegment of a Trapezoid – connects the midpoints of the legs of the trapezoid

6.21 – The midsegment of a trapezoid is parallel to each base, and its length is half the sum of the lengths of the bases.

b1

b2

2)(

2

1 2121

bbbbXY

18

12

?

?

20

27

2x + 2

3x – 1

3x + 3

302/1 – 10, 11 – 31 odd, 37 – 41odd

A B

CD

AB DC, A C