Section 5.1 - Parallelograms11/15

• Remember: a quadrilateral is any 4-sided polygon.

• Parallelograms are a type of quadrilateral with both pairs of opposite sides parallel.

• Recall: The sum of the interior angles of a parallelogram (and quadrilateral) is 360 ْ.• If you know a shape is a parallelogram, then it has 4 big properties (theorems).

• Parallelograms are named by 4 vertices, start with any and go around.

Section 5.1 - Parallelograms

A B

D C

ABCD

BCDA

CDAB

CBAD DCBA

DABC

ADCB BADC

C l o c k w i s e

C o u n t e r C l o c k w i s e

Theorem #1: The opposite sides of a parallelogram are congruent.

Section 5.1 - Parallelograms

Theorem #1: The opposite sides of a parallelogram are congruent.

Proving Theorem #1:

Section 5.1 - Parallelograms

Theorem #2: The opposite angles of a parallelogram are congruent

Section 5.1 - Parallelograms

Theorem #2: The opposite angles of a parallelogram are congruent

Proving Theorem #2:

Using the proof from Theorem #1, we know (given): andby ASA

Therefore, with CPCTC, <B = <D.And because <1 = <2 and <3 = <4, by using angle addition postulate and

substitution, we can also conclude that <A = <C.

Section 5.1 - Parallelograms

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Theorem #3: The diagonals of a parallelogram bisect each other.

Section 5.1 - Parallelograms

Theorem #3: The diagonals of a parallelogram bisect each other.

Proving Theorem #3:

Using the proof from Theorem #1 and #2,

we know (given): ΔABC = ΔCDA and ΔDBA = ΔBDC

Section 5.1 - Parallelograms

5

6 7

8

5

6 7

8

O

OO

By ASA ΔDCO = ΔBAO

By CPCTC OC = AO

By midpt theorem

AO = ½AC

therefore O is midpoint of AC

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Theorem #4: The same-side interior angles (consecutive angles) of a parallelogram are supplementary angles.

Section 5.1 - Parallelograms

A

B

C

DےA +ے B = 180 D = 180 ے+ Cےْ ْA = 180 ے+ Cے B = 180 ے+ Dےْ ْ

The following quadrilaterals are parallelograms.

Example 1

Solve for w,x,y,z.

Example 2

Solve for j,k,l,m,n.

Parallelograms – We Do

B

D

A

C115º

10

18w x

y

z

F

H

E

G

55º 65º

jk

l40ºn

m

w = 115º x = 65ºy = 10 z = 18

j = 60º k = 55º

l = 120º m = 40º

n = 20º

• Textbook practice: Pg 168 #2-8

Parallelograms – You Do