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33 Proving Lines ParallelMathematics Florida Standards
Extends MAFS.912.G-C0.3.9 Prove theorems about
lines and angles. Theorems include:... when a transversalcrosses parallel lines, alternate interior angles are
congruent and corresponding angles are congruent...
MP 1, 7
Objective To determine whether two lines are parallel
How cart you usetheorems youalready know tosolve this maze
problem?
MATHEMATICAL
PRACTICES
Lesson
Vocabularyflow proof
Getting Ready! X
The maze below has two intersecting sets of poralle! paths. A mouse makesfive turns in the maze to get to a piece of cheese. Follow the mouse's paththrough the maze. What are the number of degrees at each turn? Explainhow you know.
In the Solve It, you used parallel lines to find congruent and supplementary
relationships of special angle pairs. In this lesson, you will do the converse. You will
use the congruent and supplementary relationships of the special angle pairs to prove
lines parallel.
Essential Understanding You can use certain angle pairs to decide whether twolines are parallel.
Theorem 3-4 Converse of the Corresponding Angles Theorem
Theorem
If two lines and a transversal
form corresponding angles
that are congruent, then the
lines are parallel.
Then
€llm
You will prove Theorem 3-4 In Lesson 5-5.
156 Chapter 3 Parallel and Perpendicular Lines
A
Identifying Parallel Lines
Which line is the
transversal for z.1
and Z.2?
Line m Is the transversal
because it forms one side
of both angles.
Which lines are parallel if LI ̂ Z.2? Justify your answer.
LI and Z.2 are corresponding angles. If Z.1 = Z.2, then a
by the Converse of the Corresponding Angles Theorem.
Got It? 1. Which lines are parallelif Z.6 = Z.7? Justifyyour answer.
You can use the Converse of the Corresponding Angles Theorem to prove the following
converses of the theorems and postulate you learned in Lesson 3-2.
Theorem 3-5 Converse of the Alternate Interior Angles Theorem
Theorem
If two lines and a
transversal form alternate
interior angles that are
congruent, then the two
lines are parallel.
Then ...
e\\m
Theorem 3-6 Converse of the Same-Side Interior Angles Postulate
Theorem
If two lines and a
transversal form same-
side interior angles that
are supplementary, then
the two lines are parallel.
If...
mZ3 + mL& = 180
Then ...
€llm
Theorem 3-7 Converse of the Alternate Exterior Angles Theorem
Theorem
If two lines and a
transversal form alternate
exterior angles that are
congruent, then the two
lines are parallel.
Then ...
€ II m
C PowerGeometry.com Lesson 3-3 Proving Lines Parallel 157
The proof of the Converse of the Alternate Interior Angles Theorem below looks
different than any proof you have seen so far in this course. You know two forms of
proof—paragraph and two-column. In a third form, called flow proof, arrows show the
logical connections between the statements. Reasons are written below the statements.
Proof Proof of Theorem 3-5: Converse of the Alternate Interior Angles Theorem
Given: z.4 = z.6
Prove: i II m
^6
Given
Vertical A are
Transitive
Property of =
€ II m
If corresp. A are sthen the lines are
Proof
Given: A1 = A7
Prove: f || m
Writing a Flow Proof of Theorem 3-7
•Xir^ow
• A^ = A7
From the diagram you know• Z.1 and Z.3 are vertical
• Z.5and Z.7 are vertical
• Z.1 and Z.5 are corresponding• Z.3 and Z.7 are corresponding
One pair ofcorresponding anglescongruent to provedim
Psicin
Use a pair of congruentvertical angles to relateeither Z. 1 or Z. 7 to its
corresponding angle.
z:l s z.7
Given
^3 = ̂1
Vertical A are
A3 = A7
Transitive
Property of =
€||m
If corresp. A are s
then the lines are
Got It? 2, Use the same diagram from Problem 2 to Prove Theorem 3-6,Given: mZ.3 -I- mZ.6 = 180
Prove: II m
158 Chapters Parallel and Perpendicular Lines
T' •How do Z.1 and Z.2
relate to each other
in the diagram?Z.1 and ̂ 2 are both
exterior angles and theylie on opposite sides ofthe transversal.
The four theorems you have just learned provide you with four ways to determine if two
lines are parallel.
Determining Whether Lines are Parallel
The fence gate at the right is made up of pieces of
wood arranged in various directions. Suppose
A1 s z.2. Are lines r and s parallel?
; Explain.
Yes, r II s. Z.1 and Z.2 are alternate exteriorangles. If two lines and a transversal form
congruent alternate exterior angles, then the
lines are parallel (Converse of the Alternate
Exterior Angles Theorem).
^ Go. It? 3. In Problem 3, what is another way toexplain why r || s? Justify your answer.
in
You can use algebra along with the postulates and theorems from Lesson 3-2 and
Lesson 3-3 to help you solve problems involving parallel lines.
Work backward.
Think about what must
be true of the givenangles for a and b to beparallel.
Problem 4 Using Algebra
Algebra What is the value of x for which a jj h?
The two angles are same-side interior angles. By the Converse
of the Same-Side Interior Angles Postulate, a || if the anglesare supplementary.
Def. of supplementary angles
Simplify.
Subtract 120 from each side.
Divide each side by 2.
(2a:+ 9)+ 111 = 180
2x + 120 = 180
2x = 60
x = 30
Got It? 4. What is the value of w for which c || rf?
3w - 2
2x + 9
C PowerGeometry.com I Lesson 3-3 Proving Lines Parallel 159
Lesson Check
Do you know HOW?
State the theorem or postulate that proves a || b.
1. a
2.
Do you UNDERSTAND? ̂ ^PRACIICE^S4. Explain how you know when to use the Alternate
Interior Angles Theorem and when to use the
Converse of the Alternate Interior Angles Theorem.
5. Compare and Contrast How are flow proofs andtwo-column proofs alike? How are they different?
6. Error Analysis A classmate ^
says that AB || DC based on thediagram at the right. Explain your
classmate's error. 183'= 97=
3. What is the value ofy for which a || b in Exercise 2?J
Practice and Problem-Solving Exercises
Practice Which lines or segments are parallel? Justify your answer.
MATHEMATICAL
PRACTICES
^ See Problem 1.
7.
9. C
M A R
11. Developing Proof Complete the flow proof below.
Given: /LI and /.S are supplementary.
Prove: a II h
^ See Problem 2.
Z.1 and AS are
supplementary.
a.
d._?_
Supplements of the
same z are s.
a II fa
e. ?
b._L
Def. of linear pair
Z.1 and Z.2 are
supplementary.
c. ?
160 Chapter 3 Parallel and Perpendicular Lines
12. Parking Two workers paint lines for angled
parking spaces. One worker paints a line so
that mZ.1 = 65. The other worker paints a
line so that ml.2 = 65. Are their lines parallel?
Explain.
Algebra Find the value of j; for which C || m.
13. /^ /55°
y /(X + 25)° ^
3x - 33
{2x + 26)
Apply Developing Proof Use the given information to determine which lines,
if any, are parallel. Justify each conclusion with a theorem or postulate.
17. Z.2 is supplementary to Z.3.
19. Z.6 is supplementary to Z.7.
21. m.i7 = 65, mZ.9=115
23. ̂ 1 = Z.8
25. Zll = ̂7
Algebra Find the value of x for which C || m.
27. ^ if im
18. Z.1 = Z.3
20. = Z.12
22. Z.2 = ̂10
24. Z.8 = ̂6
26. Z.5 = Z.10
(5x + 40}
\
^ See Problem 3.
^ See Problem 4.
9/1110/12
29. Write a paragraph proof.rtSof
Given: Z.2 is supplementary to Z7.
Prove: € II m
C PowerGeometry.com I Lesson 3-3 Proving Lines Parallel 161
30. Think About a Plan If the rowing crew at the rightstrokes in unison, the oars sweep out angles of equal
measure. Explain why the oars on each side of the
shell stay parallel.
• What type of information do you need to prove
lines parallel?
• How do the positions of the angles of equal
measure relate?
Algebra Determine the value ofa; for which r || s.Then find mZ.1 and mz.2.
31. m/.l = 80 - jc, = 90 - 2a:
32. mZ.1 = 60 - 2a:, m^2 = 70 - 4a:
33. mZ.1 = 40 - 4a:, mZ.2 = 50 - Bx
34. mZl = 20 - 8x, mA2 = 30 - 16x
Use the diagram at the right below for Exercises 35-41.
?ib
Open-Ended Use the given information. State another fact about one of the
given angles that will guarantee two lines are parallel. Tell which lines will be
parallel and why.
35. Z1 = Z3
37. s /Lll
36. mZ.8 = 110, mZ.9 — 70
38. Z.I1 and Z.12 are supplementary.
39. Reasoning Ifz.1 = Z.7, what theorem or postulate can you use to
show that € II n?
Write a flow proof.
40. Given: i || n, Z.12 = Z.8Prove: j\\k
Given: j || k, mZL8 + mZ.9 = 180Prove: e 1! n
Challenge Which sides of quadrilateral PLAN must be parallel? Explain.
42. m^P = 72, mZL = 108, m/.A = 72, mAN = 108
43. mAP = 59, m^L = 37, m^A = 143, m/LN= 121
44. mAP = 67, ml.L = 120, mZJ\. = 73, mAN = 100
45. mAP = 56, mAL = 124, mAA = 124, mAN = 56
46. Write a two-column proof to prove the following: If a transversal intersects twoProof parallel lines, then the bisectors of two corresponding angles are parallel. {Hint:
Start by drawing and marking a diagram.)
162 Chapter 3 Parallel and Perpendicular Lines
SAT/Aa
Standardized Test Prep
Use the diagram for Exercises 47 and 48.
47. For what value of x is c || rf?
21 CO 43
CO 23 CO 53
48. If c II d, what is m/Lll
CO 24 CO 136
CO 44 CO 146
49. Which of the following is always a valid conclusion for the hypothesis?
If two angles are congruent, then ? .
CO they are right angles CO they have the same measure
CO they share a vertex CO they are acute angles
A
x + 21
Short
. Response
50. What is the value of x in the diagram at tlie right?
CO 1-6 CO 17
CO 10 CO 19 "*~"
51. Draw a pentagon. Is your pentagon convex or concave? Explain.
{8x-10r
Mixed Review
Find mLl and mLZ. Justify each answer.
52.
^ See Lesson 3-2.
D
Get Ready! To prepare for Lesson 3-4, do Exercises 54-57.
Determine whether each statement is always, sometimes, or never true.
54. Perpendicular lines meet at right angles.
55. Two lines in intersecting planes are perpendicular.
56. Two lines in the same plane are parallel.
57. Two lines in parallel planes are perpendicular.
♦ See Lessons 1-6 and 3-1.
I Lesson 3-3 Proving Lines Parallel 163