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3.3: Proving Lines Parallel
Learning Objective :
• SWBAT use the angles formed by a transversal
to prove two lines are parallel.
• Converse
Converse
– Think of it as the opposite.
– Before we learned that if parallel lines are cut by a
transversal, then it creates corresponding, alternate
interior, alternate exterior, and same-side interior.
– Converse means the opposite, today we will show that if
two lines are cut by a transversal and have congruent
angles (Corresponding, alternate interior, alternate
exterior) OR supplementary angles (Same-Side interior),
then the lines are parallel.
Proving Lines are Parallel
Example 1:
Use the Converse of the Corresponding Angles Postulate and
the given information to show that ℓ || m.
4  8
4  8
ℓ || m
4 and 8 are corresponding angles.
Conv. of Corr. s Post.
Example 2
Use the Converse of the Corresponding Angles Postulate and
the given information to show that ℓ || m.
m3 = (4x – 80)°,
m7 = (3x – 50)°, x = 30
m3 = 4(30) – 80 = 40 Substitute 30 for x.
m8 = 3(30) – 50 = 40 Substitute 30 for x.
m3 = m8
Trans. Prop. of Equality
3  8
Def. of  s.
ℓ || m
Conv. of Corr. s Post.
Example 3: Determining Whether Lines are Parallel
Use the given information and the theorems you have learned to
show that r || s.
4  8
4  8 4 and 8 are alternate exterior angles.
r || s
Conv. Of Alt. Int. s Thm.
Example 4: Determining Whether Lines are Parallel
Use the given information and the theorems you have learned to
show that r || s.
m2 = (10x + 8)°,
m3 = (25x – 3)°, x = 5
Example 5
Show that r || s.
m3 = 2x, m7 = (x + 50),
x = 50
Example 6: Carpentry Application
A carpenter is creating a woodwork pattern and wants two
long pieces to be parallel. m1= (8x + 20)° and m2 = (2x +
10)°. If x = 15, show that pieces A and B are parallel.
A line through the center of the horizontal piece forms a
transversal to pieces A and B.
1 and 2 are __________. If 1 and 2
are ___________, then pieces A and B are
_____________.
Substitute 15 for x in each expression.
Example 6: Carpentry Application
A carpenter is creating a woodwork pattern and wants two
long pieces to be parallel. m1= (8x + 20)° and m2 = (2x +
10)°. If x = 15, show that pieces A and B are parallel.
Short Quiz!
Lesson Quiz: Part II
Use the theorems and given information to prove p || r.
5. m2 = (5x + 20)°, m 7 = (7x + 8)°, and x = 6
m2 = 5(6) + 20 = 50°
m7 = 7(6) + 8 = 50°
m2 = m7, so 2 ≅ 7
p || r by the Conv. of Alt. Ext. s Thm.