6-2 Properties of Parallelograms page 294

Objective: To use relationships among sides, angles, diagonals or

transversals of parallelograms.

Vocabulary

Consecutive angles – angles of a polygon that share a side.

NOTE: Consecutive angles of a parallelogram are supplementary.

A B

CD

You can use what you know about parallel lines & transversals to prove some theorems about parallelograms

Theorem 6.1 p. 294---Opposite sides of a parallelogram are congruent

Theorem 6-1

Opposite sides of a parallelogram are congruent.

AB = DC

AD = BC

A B

CD

Use KMOQ to find m O.

Q and O are consecutive angles of KMOQ, so they are supplementary.

Definition of supplementary anglesm O + m Q = 180

Substitute 35 for m Q.m O + 35 = 180

Subtract 35 from each side.m O = 145

Properties of Parallelograms

6-2

Theorem 6-2

Opposite angle of a parallelogram are congruent.

<A = <C

<B = <D

A B

CD

Find the value of x in ABCD. Then find m A.

2x + 15 = 135 Add x to each side.

2x = 120 Subtract 15 from each side.

x = 60 Divide each side by 2.

x + 15 = 135 – x Opposite angles of a are congruent.

Substitute 60 for x. m B = 60 + 15 = 75

Consecutive angles of a parallelogram are supplementary.

m A + m B = 180

Subtract 75 from each side.m A = 105

m A + 75 = 180 Substitute 75 for m B.

6-2

Theorem 6-3

The diagonals of a parallelogram bisect each other.

Find the values of x and y in KLMN.

x = 7y – 16 The diagonals of a parallelogram bisect each other.2x + 5 = 5y

2(7y – 16) + 5 = 5y Substitute 7y – 16 for x in the second equation to solve for y.

14y – 32 + 5 = 5y Distribute.

14y – 27 = 5y Simplify.

Properties of Parallelograms

–27 = –9y Subtract 14y from each side.

3 = y Divide each side by –9.

x = 7(3) – 16 Substitute 3 for y in the first equation to solve for x.

x = 5 Simplify.So x = 5 and y = 3.

6-2

Theorem 6-4

If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

BD = DFA B

C D

E F

Closure

Lesson 6-1 defined a rectangle as a parallelogram with four right angles. Explain why you can now define a rectangle as a parallelogram with one right angle.

Summary

What is true about the opposite sides of a parallelogram?

What is true about the opposite angles of a parallelogram? What about consecutive angles?

What about the diagonals of a parallelogram?

When 3 or more parallel lines cut of congruent segments on one transversal, what is true about all other transversals?

Assignment 6.2

Page 297#2-32 E, 34, 35, 39-41