4.3 Proving Lines are Parallel

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Name
Class
4.3
Date
Proving Lines are Parallel
Essential Question: How can you prove that two lines are parallel?
Explore
Resource
Locker
Writing Converses of Parallel Line Theorems
You form the converse of and if-then statement "if p, then q" by swapping p and q.
The converses of the postulate and theorems you have learned about lines cut by a transversal
are true statements. In the Explore, you will write specific cases of each of these converses.
The diagram shows two lines cut by a transversal t. Use the diagram and the given statements
in Steps A–D. You will complete the statements based on your work in Steps A–D.
t
1 2
4 3
Statements
lines ℓ and m are parallel
∠2 ≅ ∠
∠2 and ∠
∠
A
are supplementary
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then
lines ℓ and m are parallel
∠2 and ∠3 are supplementary
lines ℓ and m are parallel
, then ∠2 and ∠ 3
,
.
Repeat to illustrate the Alternate Interior Angles Theorem and its converse using the
diagram and the given statements.
By the theorem: If
By its converse: If
then
D
≅ ∠7
Now write the converse of the Same-Side Interior Angles Postulate using the diagram and
your statement in Step A.
By its converse: If
C
m
Use two of the given statements together to complete a statement about the diagram using
the Same-Side Interior Angles Postulate.
By the postulate: If
are supplementary.
B
56
8 7
ℓ
lines ℓ and m are parallel
∠2 ≅ ∠6
lines ℓ and m are parallel
, then ∠2 ≅ ∠ 6 .
,
.
Use the diagram and the given statements to illustrate the Corresponding Angles Theorem
and its converse.
By the theorem: If
lines ℓ and m are parallel
By its converse: if ∠5 ≅ ∠7, then lines ℓ and m are parallel .
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, then ∠ 5
≅ ∠7.
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Reflect
1.
How do you form the converse of a statement?
Possible answer: Reverse the hypothesis and conclusion; for a statement “ifp, then q”, the
converse is “if q, then p.”
2.
What kind of angles are ∠2 and ∠6 in Step C? What does the converse you wrote in Step C mean?
Possible answer: alternate interior angles; if the two alternate interior angles ∠2 and ∠3
are congruent, then the lines ℓ and m are parallel.
Explain 1
Proving that Two Lines are Parallel
The converses from the Explore can be stated formally as a postulate and two theorems.
(You will prove the converses of the theorems in the exercises.)
Converse of the Same-Side Interior Angles Postulate
If two lines are cut by a transversal so that a pair of same-side interior angles are
supplementary, then the lines are parallel.
Converse of the Alternate Interior Angles Theorem
If two lines are cut by a transversal so that any pair of alternate interior angles are
congruent, then the lines are parallel.
Converse of the Corresponding Angles Theorem
If two lines are cut by a transversal so that any pair of corresponding angles are
congruent, then the lines are parallel.
You can use these converses to decide whether two lines are parallel.
to make an ornamental design. Each tile is congruent to the
one shown here.
120°
60°
60°
© Houghton Mifflin Harcourt Publishing Company
Example 1 A mosaic designer is using quadrilateral-shaped colored tiles
120°
ℓ1
The designer uses the colored tiles to create the pattern shown here.
ℓ2
 Use the values of the marked angles to show that the two lines
ℓ1
1
ℓ 1 andℓ 2 are parallel.
Measure of
ℓ2
∠1: 120° Measure of ∠2: 60°
2
Relationship between the two angles: They are supplementary.
Conclusion: ℓ 1 ǁ ℓ 2 by the Converse of the Same-Side Interior Angles Postulate.
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B
Now look at this situation. Use the values of the marked
angles to show that the two lines are parallel.
120°
Measure of ∠1:
120°
Measure of ∠2:
ℓ1
1
Relationship between the two
angles: They are congruent corresponding angles.
ℓ2
2
Conclusion:
ℓ 1 ǁ ℓ 2 by the Converse of the Corresponding Angles Theorem.
Reflect
3.
What If? Suppose the designer had been working with this basic shape instead. Do you
think the conclusions in Parts A and B would have been different? Why or why not?
110° 70°
70°
110°
No, because the tile pattern formed still has congruent corresponding angle and
supplementary angle pairs that can be used to produce parallel lines.
Your Turn
Explain why the lines are parallel given the angles shown. Assume that all tile
patterns use this basic shape.
4.
ℓ1
5.
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60°
120°
ℓ1
1
1
ℓ2
60°
120°
ℓ2
2
2
m∠1 = 120° and m∠2 = 120°
m ∠1 = 120° and m∠2 = 60°
They are congruent alternate interior
The angles are supplementary. The lines
angles. The lines are parallel because of
are parallel because of the Converse of
the Converse of the Alternate Interior
the Same-Side Interior Angles Postulate.
Angles Theorem.
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Explain 2
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Constructing Parallel Lines
The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel
line through a point that is not on ℓ.
The Parallel Postulate
Through a point P not on line ℓ, there is exactly one line parallel to ℓ.
Example 2 Use a compass and straightedge to construct parallel lines.
 Construct a line
Step 1
m through a point P not on a line ℓ so that m is parallel to ℓ.
Draw a line ℓ and a point P not on ℓ.
P
ℓ
Step 2
Choose two points on ℓ and label them Q and R. Use a
‹ ›
−
straightedge to draw PQ .
P
ℓ
Q
R
Step 3
Use a compass to copy ∠PQR at point P, as shown, to
construct line m.
P
m
 In the space provided, follow the steps to construct a line
ℓ
Q
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line m ǁ line ℓ
R
r through
a point G not on a line s so that r is parallel to s.
G
r
S
Step 1
Step 2
Step 3
E
F
Draw a line s and a point G not on s.
‹ ›
−
Choose two points on s and label them E and F. Use a straightedge to draw GE .
Use a compass to copy ∠GEF at point G. Label the side of the angle as line r.
line r ǁ line s
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Reflect
6.
Discussion Explain how you know that the construction in Part A or Part B produces a line passing
through the given point that is parallel to the given line.
In each case, the construction creates two congruent corresponding angles. In Part A,
for example, lines ℓ and m are cut by a transversal and a pair of corresponding angles
are congruent, so the lines are parallel (Converse of the Corresponding Angles Theorem).
Your Turn
7.
ℓ
Construct a line m through P parallel to line ℓ.
m
P
Using Angle Pair Relationships to Verify Lines
are Parallel
Explain 3
When two lines are cut by a transversal, you can use relationships of pairs of angles to decide if the lines are parallel.
Example 3 Use the given angle relationships to decide whether the
t
lines are parallel. Explain your reasoning.
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
ℓ
∠3 ≅ ∠5
Step 1
Identify the relationship between the two angles.
∠3 and ∠5 are congruent alternate interior angles.
Step 2
Are the lines parallel? Explain.
Yes, the lines are parallel by the Converse of the Alternate
Interior Angles Theorem.

m
1 2
4 3
5 6
8 7
m∠4 = (x + 20)°,m ∠8 = (2x + 5)°,and x = 15.
Step 1
Identify the relationship between the two angles.
m∠4 = (x + 20)°
=
So,
Step 2
( 15
∠4
and
)
m∠8 = (2x + 5)°
(
°
+ 20 = 35°
∠8
= 2⋅
)
°
15 + 5 = 35°
are congruent corresponding angles.
Are the lines parallel? Explain.
Yes, the lines are parallel by the Converse of the Corresponding Angles Theorem.
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Your Turn
t
Identify the type of angle pair described in the given
condition. How do you know that lines ℓ and m are parallel?
1 2
4 3
ℓ
m∠3 + m∠6 = 180°
same side interior angles; by the Converse of the Same Side
8.
Interior Angles Postulate
5 6
8 7
m
∠2 ≅ ∠6
corresponding angles; by the Converse of the Corresponding
9.
Angles Theorem
Elaborate
10. How are the converses in this lesson different from the postulate/theorems in the previous lesson?
In the previous lesson, we knew lines were parallel and things about angles; here, we
know things about angle pairs, and lines are parallel.
11. What If? Suppose two lines are cut by a transversal such that alternate interior angles are both congruent
and supplementary. Describe the lines.
The lines are parallel and all the angles are 90°. The transversal is perpendicular to the
lines.
12. Essential Question Check-In Name two ways to test if a pair of lines is parallel, using the interior
angles formed by a transversal crossing the two lines.
Possible answer: Use given information or measure pairs of angles to decide if alternate
interior angles are congruent or if same-side interior angles are supplementary.
• Online Homework
• Hints and Help
• Extra Practice
The diagram shows two lines cut by a transversal t. Use the diagram
and the given statements in Exercises 1–3 on the facing page.
Statements
t
1 2
4 3
lines ℓ and m are parallel
+ m∠7 = 180°
m∠
56
8 7
∠1 ≅ ∠
∠
ℓ
m
© Houghton Mifflin Harcourt Publishing Company
Evaluate: Homework and Practice
≅ ∠6
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