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3-3
3-5 Proving
ProvingLines
LinesParallel
Parallel
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
3-5 Proving Lines Parallel
3-3
Warm Up
State the converse of each statement.
1. If a = b, then a + c = b + c.
If a + c = b + c, then a = b.
2. If mA + mB =
complementary.
If A and  B are
then mA + mB
3. If AB + BC = AC,
90°, then A and B are
complementary,
=90°.
then A, B, and C are collinear.
If A, B, and C are collinear, then AB + BC = AC.
Holt Geometry
3-5 Proving Lines Parallel
3-3
Objective
Use the angles formed by a transversal
to prove two lines are parallel.
Holt Geometry
3-3
3-5 Proving Lines Parallel
What does x equal?
100
x
Holt Geometry
We don’t know
because we have no
idea if the lines are
parallel
3-5 Proving Lines Parallel
3-3
What does x equal?
100
x
Holt Geometry
Since the lines are
parallel, x=100.
3-5 Proving Lines Parallel
3-3
100
What do you know
about the lines?
They are not parallel.
25
Holt Geometry
3-5 Proving Lines Parallel
3-3
100
100
Holt Geometry
What do you know
about the lines?
The lines must be
parallel.
3-5 Proving Lines Parallel
3-3
Make an “if…then” statement about each figure’s angles and lines.
If two parallel lines are
cut by a transversal,
then the corresponding
angles are congruent.
100
100
Holt Geometry
Corresponding angles Converse:
If two lines are cut by a
transversal so that the
corresponding angles are
congruent, then the lines are
parallel.
3-5 Proving Lines Parallel
3-3
Make an “if…then” statement about each figure’s angles and lines.
If two parallel lines are cut by
a transversal, then the
alternate interior angles are
congruent.
50
50
Holt Geometry
Alt. Interior Angles Converse:
If two lines are cut by a
transversal so that the
alternate interior angles are
congruent, then the lines are
parallel.
3-5 Proving Lines Parallel
3-3
Make an “if…then” statement about each figure’s angles and lines.
If two parallel lines are cut by
a transversal, then the
alternate exterior angles are
congruent.
60
60
Holt Geometry
Alt. Exterior Angles Converse:
If two lines are cut by a
transversal so that the
alternate exterior angles are
congruent, then the lines are
parallel.
3-5 Proving Lines Parallel
3-3
Make an “if…then” statement about each figure’s angles and lines.
If two parallel lines are cut by
a transversal, then the
consecutive interior angles are
supplementary.
80
100
Holt Geometry
Consecutive Angles Converse:
If two lines are cut by a
transversal so that the
consecutive interior angles are
supplementary, then the lines
are parallel.
3-5 Proving Lines Parallel
3-3
Example 1: Using the Converse of the Corresponding
Angles Postulate
A) Use the Converse of the Corresponding
Angles Postulate and the given information to
show that ℓ || m.
4  8
Yes, ℓ || m because of the Conv. of Corr. s Post.
Holt Geometry
3-5 Proving Lines Parallel
3-3
B) Use the Converse of the Corresponding
Angles Postulate and the given information to
show that ℓ || m. (aka plug in the value of x
and see if it gives you a true statement.)
m3 = (4x – 80)°,
m7 = (3x – 50)°, x = 30
m3 = 4(30) – 80 = 40
m7 = 3(30) – 50 = 40
3  7
Yes, ℓ || m because of the Conv. of Corr. s Post.
Holt Geometry
3-5 Proving Lines Parallel
3-3
C) Use the Converse of the Alt. Int. Angles
Postulate and the given information to show
that ℓ || m.
m7 = (4x + 26)°,
m2 = (5x + 12)°, x = 13
m7 = 4(13) + 26 = 78
m2 = 5(13) + 12 = 77
Substitute 13 for x.
Substitute 13 for x.
m7 = m2
ℓ is not parallel to m because alternate interior angles
should be congruent, and these two angles are not.
Holt Geometry
3-5 Proving Lines Parallel
3-3
D) Use the Converse of the Alt. Ext. Angles
Postulate and the given information to show
that ℓ || m.
m1 = (3x + 10)°,
m8 = (4x - 10)°, x = 20
m1 = 3(20) + 10 = 70
Substitute 13 for x.
m8 = 4(20) - 10 = 70Substitute 13 for x.
m1 = m8
Yes, ℓ || m because of the converse of
alternate exterior angles
Holt Geometry
3-5 Proving Lines Parallel
3-3
E) Use the given information and the theorems
you have learned to show that r || s.
m2 = (10x + 8)°,
m3 = (25x – 67)°, x = 5
m2 = 10x + 8
= 10(5) + 8 = 58
m2 + m3 = 58° + 58°
= 116°
m3 = 25x – 67
= 25(5) – 67 = 58
2 and 3 are same-side
interior angles.
So, r is not parallel to s because the angles should =180
Holt Geometry
3-5 Proving Lines Parallel
3-3
Example 4: 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.
Holt Geometry
3-5 Proving Lines Parallel
3-3
Example 4 Continued
m1 = 8x + 20
= 8(15) + 20 = 140
Substitute 15 for x.
m2 = 2x + 10
= 2(15) + 10 = 40
m1+m2 = 140 + 40
= 180
Substitute 15 for x.
1 and 2 are
supplementary.
The same-side interior angles are supplementary, so
pieces A and B are parallel by the Converse of the
Same-Side Interior Angles Theorem.
Holt Geometry
3-5 Proving Lines Parallel
3-3
Check It Out! Example 4
What if…? Suppose the
corresponding angles on
the opposite side of the
boat measure (4y – 2)°
and (3y + 6)°, where
y = 8. Show that the oars
are parallel.
4y – 2 = 4(8) – 2 = 30°
3y + 6 = 3(8) + 6 =
30°
The angles are congruent, so the oars are || by the
Conv. of the Corr. s Post.
Holt Geometry