Download Proofs with Parallel Lines 3.3 and more Exercises Reasoning in PDF only on Docsity!
Section 3.3 Proofs with Parallel Lines 137
3.3 Proofs with Parallel Lines
Exploring Converses
Work with a partner. Write the converse of each conditional statement. Draw a diagram to represent the converse. Determine whether the converse is true. Justify your conclusion. a. Corresponding Angles Theorem (Theorem 3.1) If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Converse
b. Alternate Interior Angles Theorem (Theorem 3.2) If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Converse
c. Alternate Exterior Angles Theorem (Theorem 3.3) If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Converse
d. Consecutive Interior Angles Theorem (Theorem 3.4) If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Converse
Communicate Your AnswerCommunicate Your Answer
2. For which of the theorems involving parallel lines and transversals is the converse true? 3. In Exploration 1, explain how you would prove any of the theorems that you found to be true.
CONSTRUCTING
VIABLE
ARGUMENTS
To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.
Essential QuestionEssential Question For which of the theorems involving parallel
lines and transversals is the converse true?
1 4
2 3
6 8 7
5
1 4
2 3
6 8 7
5
1 4
2 3
6 8 7
5
1 4
2 3
6 8 7
5
138 Chapter 3 Parallel and Perpendicular Lines
3.3 Lesson^ What You Will LearnWhat You Will Learn
Use the Corresponding Angles Converse. Construct parallel lines. Prove theorems about parallel lines. Use the Transitive Property of Parallel Lines.
Using the Corresponding Angles Converse Theorem 3.5 below is the converse of the Corresponding Angles Theorem (Theorem 3.1). Similarly, the other theorems about angles formed when parallel lines are cut by a transversal have true converses. Remember that the converse of a true conditional statement is not necessarily true, so you must prove each converse of a theorem.
Previous converse parallel lines transversal corresponding angles congruent alternate interior angles alternate exterior angles consecutive interior angles
Core VocabularyCore Vocabullarry
Using the Corresponding Angles Converse
Find the value of x that makes m n.
SOLUTION
Lines m and n are parallel when the marked corresponding angles are congruent.
(3 x + 5)° = 65 ° Use the Corresponding Angles Converse to write an equation.
3 x = 60 Subtract 5 from each side.
x = 20 Divide each side by 3.
So, lines m and n are parallel when x = 20.
Monitoring ProgressMonitoring Progress (^) Help in English and Spanish at BigIdeasMath.com
1. Is there enough information in the diagram to conclude that m n? Explain.
n
m
75 °
105 °
2. Explain why the Corresponding Angles Converse is the converse of the Corresponding Angles Theorem (Theorem 3.1).
TheoremTheorem
Theorem 3.5 Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
Proof Ex. 36, p. 180
j
k
6
2
m
n
65 °
(3 x + 5)°
j ^ k
140 Chapter 3 Parallel and Perpendicular Lines
Proving Theorems about Parallel Lines
Proving the Alternate Interior Angles Converse
Prove that if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
SOLUTION
Given ∠ 4 ≅ ∠ 5 Prove g h
STATEMENTS REASONS
1. ∠ 4 ≅ ∠ 5 1. Given 2. ∠ 1 ≅ ∠ 4 2. Vertical Angles Congruence Theorem (Theorem 2.6) 3. ∠ 1 ≅ ∠ 5 3. Transitive Property of Congruence (Theorem 2.2) 4. g h 4. Corresponding Angles Converse
Determining Whether Lines Are Parallel
In the diagram, r s and ∠1 is congruent to ∠3. Prove p q.
SOLUTION
Look at the diagram to make a plan. The diagram suggests that you look at angles 1, 2, and 3. Also, you may find it helpful to focus on one pair of lines and one transversal at a time. Plan for Proof a. Look at ∠1 and ∠2. ∠ 1 ≅ ∠2 because r s. b. Look at ∠2 and ∠3. If ∠ 2 ≅ ∠3, then p q. Plan for Action a. It is given that r s , so by the Corresponding Angles Theorem (Theorem 3.1), ∠ 1 ≅ ∠2. b. It is also given that ∠ 1 ≅ ∠3. Then ∠ 2 ≅ ∠3 by the Transitive Property of Congruence (Theorem 2.2). So, by the Alternate Interior Angles Converse, p q.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
3. If you use the diagram below to prove the Alternate Exterior Angles Converse, what Given and Prove statements would you use?
k
j
1
8
4. Copy and complete the following paragraph proof of the Alternate Interior Angles Converse using the diagram in Example 2. It is given that ∠ 4 ≅ ∠5. By the ______, ∠ 1 ≅ ∠4. Then by the Transitive Property of Congruence (Theorem 2.2), ______. So, by the ______, g h.
h
4 g 5
1
2 1
p
r s
q
3
Section 3.3 Proofs with Parallel Lines 141
Using the Transitive Property of Parallel Lines
The flag of the United States has 13 alternating red and white stripes. Each stripe is parallel to the stripe immediately below it. Explain why the top stripe is parallel to the bottom stripe.
s 1 s 3 s 5 s 7 s 9 s 11 s 13
s 2 s 4 s 6 s 8 s 10 s 12
SOLUTION
You can name the stripes from top to bottom as s l , s 2 , s 3 ,... , s 13. Each stripe is parallel to the one immediately below it, so s 1 s 2 , s 2 s 3 , and so on. Then s 1 s 3 by the Transitive Property of Parallel Lines. Similarly, because s 3 s 4 , it follows that s 1 s 4. By continuing this reasoning, s 1 s 13.
So, the top stripe is parallel to the bottom stripe.
Monitoring ProgressMonitoring Progress (^) Help in English and Spanish at BigIdeasMath.com
5. Each step is parallel to the step immediately above it. The bottom step is parallel to the ground. Explain why the top step is parallel to the ground. 6. In the diagram below, p q and q r. Find m ∠8. Explain your reasoning.
q
r
p
s 115 °
8
Using the Transitive Property of Parallel Lines
TheoremTheorem
Theorem 3.9 Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other.
Proof Ex. 39, p. 144; Ex. 48, p. 162
p q r
If p ^ q and q ^ r , then p ^ r.
Section 3.3 Proofs with Parallel Lines 143
In Exercises 21–24, are ⃖ AC ⃗^ and^ ⃖ DF ⃗^ parallel? Explain your reasoning.
21. A B
C
D
E F
57 ° 123 °
A B
D
C
E F
143 °
37 °
A B
D
C
E F
62 ° 62 °
A B C
D E F
65 ° 65 °
115 °
25. ANALYZING RELATIONSHIPS The map shows part of Denver, Colorado. Use the markings on the map. Are the numbered streets parallel to one another? Explain your reasoning. (See Example 4.)
E. 20th Ave.
E. 19th Ave.
E. 18th Ave.
E. 17th Ave.
Pennsylvania St.Pearl St.Washington St.Clarkson St.Ogden St.Downing St.
Park Ave. Franklin St.
26. ANALYZING RELATIONSHIPS Each rung of the ladder is parallel to the rung directly above it. Explain why the top rung is parallel to the bottom rung. 27. MODELING WITH MATHEMATICS The diagram of the control bar of the kite shows the angles formed between the control bar and the kite lines. How do you know that n is parallel to m?
n
m
108 °
108 °
28. REASONING Use the diagram. Which rays are parallel? Which rays are not parallel? Explain your reasoning.
62º
58º 59º
61º
A B C^ D
F E H G
29. ATTENDING TO PRECISION Use the diagram. Which theorems allow you to conclude that m n? Select all that apply. Explain your reasoning.
m
n
A Corresponding Angles Converse (Thm. 3.5)
B Alternate Interior Angles Converse (Thm. 3.6)
C Alternate Exterior Angles Converse (Thm. 3.7)
D Consecutive Interior Angles Converse (Thm. 3.8)
30. MODELING WITH MATHEMATICS One way to build stairs is to attach triangular blocks to an angled support, as shown. The sides of the angled support are parallel. If the support makes a 32° angle with the floor, what must m ∠1 be so the top of the step will be parallel to the floor? Explain your reasoning.
2
1
32 °
triangular block
31. ABSTRACT REASONING In the diagram, how many angles must be given to determine whether j k? Give four examples that would allow you to conclude that j k using the theorems from this lesson.
12 5
j k
34 t
6 78
144 Chapter 3 Parallel and Perpendicular Lines
32. THOUGHT PROVOKING Draw a diagram of at least two lines cut by at least one transversal. Mark your diagram so that it cannot be proven that any lines are parallel. Then explain how your diagram would need to change in order to prove that lines are parallel.
PROOF In Exercises 33–36, write a proof.
33. Given m ∠ 1 = 115 °, m ∠ 2 = 65 ° Prove m n
(^1) m 2 n
34. Given ∠1 and ∠3 are supplementary.
Prove m n
m
1 2 3 n
35. Given ∠ 1 ≅ ∠2, ∠ 3 ≅ ∠ 4 Prove AB — CD —
A
B C
D E
1 2 3 4
36. Given a b , ∠ 2 ≅ ∠ 3 Prove c d
1 2
a
c d
b 3 4
37. MAKING AN ARGUMENT Your classmate decided that ⃖ AD ⃗ ⃖ BC ⃗ based on the diagram. Is your classmate correct? Explain your reasoning.
C
A B
D
38. HOW DO YOU SEE IT? Are the markings on the diagram enough to conclude that any lines are parallel? If so, which ones? If not, what other information is needed?
r
p
q 1 2 3 4
s
39. PROVING A THEOREM Use these steps to prove the Transitive Property of Parallel Lines Theorem (Theorem 3.9). a. Copy the diagram with the Transitive Property of Parallel Lines Theorem on page 141. b. Write the Given and Prove statements. c. Use the properties of angles formed by parallel lines cut by a transversal to prove the theorem. 40. MATHEMATICAL CONNECTIONS Use the diagram.
p
r s
( x + 56)°
(2 x + 2)°
( y + 7)° (3 y − 17)° q
a. Find the value of x that makes p q. b. Find the value of y that makes r s. c. Can r be parallel to s and can p be parallel to q at the same time? Explain your reasoning.
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Use the Distance Formula to find the distance between the two points. (Section 1.3)
41. (1, 3) and (−2, 9) 42. (−3, 7) and (8, −6) 43. (5, −4) and (0, 8) 44. (13, 1) and (9, −4)
Reviewing what you learned in previous grades and lessons
