3.5 Using Properties of Parallel Lines

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3.5 Using Properties of Parallel Lines

• Geometry
• Mrs. Spitz
• Fall 2005

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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.

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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.

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Ex. 1 Proving two lines are parallel

• Lines m, n and k represent three of the oars of a
  team of rowers.
• Given mn and nk. Prove mk.

1
m
2
n
3
k
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Proof

• Statements
• mn
• ?1 ? ?2
• nk
• ?2 ? ?3
• ?1 ? ?3
• mk

• Reasons
• Given
• Corresponding Angles Postulate
• Given
• Corresponding Angles Postulate
• Transitive POC
• Corresponding Angles Converse

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Parallel/Perpendicular lines Theorems

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

r
q
p
If pq and qr, then pr.
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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.

If m?p and n?p, then mn.
m
n
p
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Ex. 2 Why steps are parallel

• In 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
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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.

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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.

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Proof

• Given m?1 45, m?2 45, m?3 45, m? 45
• Prove BACD

B
1
A
2
3
D
4
C
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m?345, m?445
m?145, m?245
m?BCD m?3 m?4
m?ABC m?1 m?2
Angle Addition Postulate
Given
Angle Addition Postulate
Given
m?ABC 90
m?BCD 90
Substitution Property
Substitution Property
?ABC is a right angle.
?BCD is a right angle.
Definition of Right Angle
Definition of Right Angle
BC?CD
BA?BC
Definition of ? lines
Definition of ? lines
BA?CD
In a plane, 2 lines ?to the same line are .
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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.

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Reminder

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