3.5 Using Properties of Parallel Lines

GeometryMs. Reser

Review Parallel Lines and Transversal 3.3

Remember if two parallel lines are cut by a transversal. Then

Corresponding angles are congruentAlternate Ext angles are congruentAlternate Int angles are congruentAnd consecutive or same side int angles are supplememntary

Find the value 0f x and y given the lines are parallel!

Proving lines are parallelIf you are given two lines cut by a transversal and you want to STATE they are parallel one of the following must be true!!!! If one of these conditions does not apply you cannot say they are parallel. These are the converse statements

Corresponding angles are congruentAlternate Ext angles are congruentAlternate Int angles are congruentAnd consecutive or same side int angles are supplementaryOR if you know the equation, points or have the line on a coordinate plane and the slopes are the same you can say they are parallel!!!

YOU CANNOT SAY THEY ARE PARALLEL BECAUSE THEY DO NOT INTERSECT!!!! YOU MUST GIVE A VALID REASON

Standard/Objectives:Standard 3: Students will learn and apply

geometric concepts.Objectives:

Use properties of parallel lines in real-life situations, such as building a CD rack.Construct parallel lines using a straight edge and a compass.To understand how light bends when it passes through glass or water.

Assignment:Quiz after this section.Pg. 159 – Do constructions on this page “Copying an Angle” and “Parallel Lines.” Place in your binder under computer/lab work. In your homework, you are asked to do constructions on pg. 161 #25-28. Place these in your binder.Pgs. 160-162 #1-24, 33-36.

Ex. 1: Proving two lines are parallel

Lines m, n and k represent three of the oars of a team of rowers. Given: m║n and n║k. Prove: m║k.

m

n

1

k

2

3

Proof:Statements:

1. m║n

2. 1 ≅ 2

3. n║k

4. 2 ≅ 3

5. 1 ≅ 3

6. m║k

Reasons:

1. Given2. Corresponding

Angles Postulate3. Given4. Corresponding

Angles Postulate5. Transitive POC6. Corresponding

Angles Converse

Parallel/Perpendicular lines Theorems

Theorem 3.11: If two lines are parallel to the same line, then they are parallel to each other.

pq r

If p║q and q║r, then p║r.

Parallel/Perpendicular lines Theorems

Theorem 3.12: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

m n

p

If mp and np, then m║n.

Ex. 2: Why steps are parallelIn the diagram at the right, each step is parallel to the step immediately below it and the bottom step is parallel to the floor. Explain why the top step is parallel to the floor.

k1

k2

k3

k4

Solution

You are given that k1║ k2 and k2║ k3. By transitivity of parallel lines, k1║ k3. Since k1║ k3 and k3║ k4, it follows that k1║ k4. So the top step is parallel to the floor.

Ex. 3: Building a CD Rack

You are building a CD rack. You cut the sides, bottom, and top so that each corner is composed of two 45° angles. Prove that the top and bottom front edges of the CD rack are parallel.

Proof:

Given: m1 = 45°, m2 = 45°, m3 = 45°, m = 45°

Prove: BA║CD1

2

34

B

C

A

D

mABC = m1+ m2Angle Addition Postulate

m1=45°, m2=45°

Given

mBCD = m3+ m4

m1=45°, m2=45°

Angle Addition Postulate Given

mABC = 90°

Substitution Property

ABC is a right angle.

Definition of Right Angle

BABCDefinition of lines

mBCD = 90°Substitution Property

BCD is a right angle.

Definition of Right Angle

BCCDDefinition of lines

BACD In a plane, 2 lines to the same line are ║.

Constructions:

Do the two constructions on page 159. Label and place in your binder.Do Constructions on page 161 #25-28. Label and place in your binder.

Reminder:

Quiz 2 will be the next time we meet. To look at a similar quiz, check out page 164.