Geometry

5.3 Theorems Involving Parallel Lines

4 UsefulTheorems

Today you will learn 4 useful theorems about parallel lines. These will help you solve problems and write proofs.

Theorem

If 2 lines are parallel, then all points on one line are equidistant from the other line.

A Bl

mC D

AC = BD

Theorem

If 3 parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

A

B

C

X

Y

Z

If AB = BC

then XY = YZ

Theorem

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.

Y Z

X

M N

If M is the midpoint of XYthen N is the midpoint of XZ

Theorem

A segment that joins the midpoints of 2 sides of a triangle; 1. is parallel to the third side.2. is half as long as the third side.

Y Z

X

M N

1.MN YZ2.MN = ½YZ

Sample Problems:

A

D

C

B

P

Q

R

S

1) Find x when PQ = 3x – 9 and QR = 2x – 2.

2) Find PR and PS when PQ = 5.

3) Find RS when PS = 33.

4) Find x when PR = x + 8 and QS = 3x – 2.

5) Find QS in problem #4.

X

Y6

A B

C Z6

X

Y

5

A B

C Z6

4

4

X

Y

6

AB

C Z5

3

3

6) Name the points shown that MUST be midpoints of the sides of the large triangle.

7) Find YZ if MN = 8.

8) Find MN if YZ = 30.

9) Find YZ if MN = 2k + 3.

10) Find MN if YZ = k.

X

Y Z

M N

m XYZ 80m XNM 40m MXN

11) Find x, MN and YZ if MN = x and YZ = 3x – 25.

12) Find if and

.

X

Y Z

M N

13) Find x and y if AB = 19, BC = x + 3y, and CD = 6x – y.

A

D

C

B

P

Q

R

S

Homework

P. 180 #1-17

Let’s Try A Few Together

Practice 18 # 6 Test 18 #2, 11, 12