CHAPTER 5 SECTION 3Theorems Involving Parallel Lines

WARM-UP: THURSDAY, MARCH 27TH

State the 4 ways to prove a quadrilateral is a parallelogram.

Part 1: Parallelograms

Methods of ProofPart 2: Algebraic Connections

What values must x and y have to make the quadrilateral a parallelogram

8x - 6 3y

42

HO

MEW

OR

K: W

ED

NES

DAY, A

PR

IL 2

p. 174 #’s 1 – 7, 14, 19 -221.) Definition of parallelogram

2.) Both pairs of opposite sides 3.) One pair of sides both || and 4.) Diagonals bisect each other5.) Both pairs of opposite angles 6.) Answers vary7.) Diagonals bisect each other14.) See next slide19.) x =18, y=1420.) x=20, y=6 or -521.) x=10, y=222.) x=11, y=5

14.)

Given: Parallelogram ABCDM and N are midpoints of AB and DC

Prove: AMCN is a parallelogram

C

BMA

ND

PARALLEL LINES…WHAT DO YOU KNOW?You should know lots… I hope

TELL ME EVERYTING YOU KNOW ABOUT…

Parallel Lines Parallelograms

THEOREM 5.9:

If parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. (not just limited to 3 lines)

Given: AX || BY||CZAB BC

Prove: XY YZY

Z

X

C

B

A

APPLICATION OF THEOREM 5.9

Given: AR||BS||CT; RS ST

RS = 12, ST =

AB = 8; BC =

AC = 20; AB =

AC = 10x; BC =

S

T

R

C

B

A

THEOREM 5.10:

A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.

Given: M is the midpoint of ABMN ||BC

Prove: N is the midpoint of AC

M

A

N

B C

PROOF OF THEOREM 5.10: Given: M is the midpoint of AB

MN ||BC

Prove: N is the midpoint of ACM

A

N

B C

D

Paragraph Proof:

Then AD, MN , and BC are three parallel lines that cut off congruent segments on transversal AB. By Theorem 5.9 they also cut off congruent segments on AC. Thus AN NC and N is the midpoint of AC.

Let AD be the line through A parallel to MN.

THEOREM 5.11:

The segment that joins the midpoints of two sides of a triangle: (1) is parallel to the third side (2) is half as long as the third side

Given: M is the midpoint of ABN is the midpoint of AC

Prove: (1) MN || BC(2) MN = ½BC

M

A

N

B C

APPLICATION OF THEOREM 5.11 Given: R, S, and T are midpoints of the sides of ABC.

B

SR

CTA

MIXING IT ALL TOGETHERApplications of Theorems 5.9, 5.10, and 5.11

ALGEBRAIC CONNECTIONS: SYSTEMS PRACTICE

If AB = 15, BC = 2x – y, and CD = x + y. Calculate x and y.

D

C

B

A

H

G

F

E

ALGEBRAIC CONNECTIONS: PERIMETER

P, Q, and R are midpoints of the sides of DEF. What kind of figure is DPQR? What is the perimeter of DPQR?

F

R Q

D P E

810

12

COOL DOWN: THURSDAY, MARCH 27TH Self Test: p. 182 #’s 1 – 8

Remember: Quiz on Sections 5.1 – 5.3 Friday!!