33, Theorems About Parallel Lines

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3-3, Theorems About Parallel Lines

• Objective
• To use theorems and definitions to find the
  measures of angles

2
Review

• What are supplementary angles?
• What are adjacent angles?
• Euclids Postulate 5
• Axiom 1 Things that are equal to the same thing
  are equal to each other.
• (Think of this as substitution.)
• Axiom 3 If equals are subtracted from equals,
  the differences are equal.

3
Theorem 3.3.1

• If two lines are parallel, then the interior
  angles on the same side of the transversal are
  supplementary

t
c
180º
a b
l
180º
d
m
4
Theorem 3.3.2

• If two lines cut by a transversal are parallel,
  then the corresponding angles are equal

t
80º
x
l
y
If x 80º, then y 80º
m
5
Theorem 3.3.3

• If two lines cut by a transversal are parallel,
  then the alternate interior angles are equal

t
c
a b
l
d
m
6
Additional Theorems
congruent
congruent
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Additional Theorems
supplementary
congruent
perpendicular
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Lesson 2 Euclids Five Postulates

• 5. If two lines l and m are cut by a third line
  t, the two inside angles (a and b) together
  measure less than two right angles, then the two
  lines l and m, if extended, will meet on the same
  side of t as the two angles a and b.
• Back to Slide 1

Example
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3-4, Constructions and Problem Solving

• Objective
• -To use the converse of a theorem to construct
  parallel lines
• -To use theorems to find the measures of angles
  formed by parallel lines and transversals

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Alternate Interior Angles Postulate

• If a transversal intersects two lines so that the
  alternate interior angles are equal, then the
  lines are parallel

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Example 1 The measure of Ðb is twice the
measure of Ða. What is the measure of each angle?

• Let Ða x and Ðb
• mÐa mÐb 180º (Consecutive Interior Angles
  Theorem)
• So x 2x 180º by substitution
• 3x 180º
• x 60º
• Ða x 60º and Ðb 2x 120º

2x
a b
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Example 2 The measure of Ða is five times the
measure of Ðb. What is the measure of Ðy?

• mÐb mÐy (Corresponding Angles)
• mÐb mÐa 180º (Supplementary Angles)
• Let mÐb x and mÐa 5x
• So x 5x 180º by substitution
• 6x 180º
• x 30º
• x Ðb 30º Ðy

y
a b
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Example 3 Using the figure shown, give two ways
to find mÐy

• 1. The angle of 150º and Ðy are corresponding
  angles.
• Therefore, mÐy 150º
• 2. mÐz 30º (180º 150º) (Supplementary)
• mÐx 150º (Ðx 30º 180º ) (Supplementary)
• Therefore, Ðx and Ðy are equal (150º) (Interior
  Angles)

150º
x z y
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Exercise B
o
75º
p
q
r
s
u
v
w
x
y
z