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Geometry
3.3 Proving Lines are Parallel
Goals
Use postulates and theorems to prove
two lines parallel.
 Solve problems with parallel lines.

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Geometry 3.3 Proving Lines are Parallel
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What we’ve been doing:
Given two parallel lines cut by a transversal…
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Geometry 3.3 Proving Lines are Parallel
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Corresponding Angles Congruent
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Alternate Exterior Angles Congruent
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Alternate Interior Angles Congruent
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Same Side Interior Angles Supplementary
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Theorem:
If two parallel lines are cut by a
transversal, then corresponding
angles are congruent.
Converse:
If two lines are cut by a transversal
and corresponding angles are
congruent, then the lines are parallel.
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In other words…
Corr. s   2 lines ||
(Theorem 3.5, Corr.  Converse)
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Theorem:
If two parallel lines are cut by a
transversal, then alternate interior
angles are congruent.
Converse:
If two lines are cut by a transversal
and alternate interior angles are
congruent, then the lines are parallel.
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Geometry 3.3 Proving Lines are Parallel
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In other words…
Alt Int s   2 lines ||
(Theorem 3.6, Alternate Interior
Angles Converse)
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Geometry 3.3 Proving Lines are Parallel
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Theorem:
If two parallel lines are cut by a
transversal, then alternate Exterior
angles are congruent.
Converse:
If two lines are cut by a transversal
and alternate Exterior angles are
congruent, then the lines are parallel.
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Geometry 3.3 Proving Lines are Parallel
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In other words…
Alt Ext s   2 lines ||
(Theorem 3.7, Alternate Exterior
Angles Converse)
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Theorem:
If two parallel lines are cut by a
transversal, then same side interior
angles are supplementary.
Converse:
If two lines are cut by a transversal
and same side interior angles are
supplementary, then the lines are
parallel.
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In other words…
y°
x + y = 180°
x°
SS Int s supp  2 lines ||
(Theorem 3.8, Same Side Interior Angles
Converse)
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To show two lines parallel, show
that one of these is true:
Corresponding angles congruent.
 Alternate interior angles congruent.
 Alternate exterior angles congruent.
 Same side interior angles
supplementary.
 You only need one pair for any one of
these reasons.

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To show two lines parallel, show
that one of these is true:
Corr. s 
 Alt. Int. s 
 Alt. Ext. s 
 SS Int. s supp.
 You only need one pair for any one of
these reasons.

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Example 1
Given: m  t, n  t.
t
Prove: m || n
m
1
Not drawn to scale
2
n Obviously
Proof:
Since m  t, 1 is a right angle.
Since n  t, 2 is a right angle.
All right angles are congruent, so 1  2.
This means m || n since alt int s   2 lines ||.
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Example 2
Given: 5  6; 6  4
Prove: AD || BC
4
B
A
5
6
D
C
Proof:
If 5  6 and 6  4, then 5  4.
So AD || BC
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Transitive Prop
because of alt int s .
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Example 3
m
n
(2x + 1)°
(3x – 5)°
Find the value of
x to make m || n.
These are alt int
angles.
2x + 1 = 3x – 5
6=x
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Write a proof
In the diagram, p || q and
∠1 is supplementary to ∠2.
Prove r ||s using a proof.
Statements
Reasons
∠1 is supp to ∠2
m ∠1 + m ∠2 = 180
p || q
m∠2=m∠3
m ∠1 + m ∠ 3 = 180
∠1 and ∠ 3 are supp
r || s
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Given
Def of supp ∠’s
Given
Alt Int ∠’s Theorem
Substitution
Def of supp ∠’s
Same Side Int ∠’s Converse
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Theorem 3.9 Transitive Property of
parallel lines
If two lines are parallel to the same line,
then they are parallel to each other.
m
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n
p
If m || n and p || n,
then m || p.
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Slat 1 is
parallel to
slat 2.
Slat 2 is
parallel to
slate 3.
1 2 3
Theorem 3.9
Why is slat 1
parallel to
slat 3?
If two lines are parallel to the same line, then they
are parallel to each other.
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Geometry 3.5 Using Properties of Parallel
Lines
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