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ProvingProvingLinesLines
ParallelParallel
ProvingProvingLinesLines
ParallelParallel
3-33-3
EXAMPLE 1 Apply the Corresponding Angles Converse
ALGEBRA Find the value of x that makes m n.
SOLUTION
Lines m and n are parallel if the marked corresponding angles are congruent.
3x = 60
x = 20
The lines m and n are parallel when x = 20.
(3x + 5)o = 65o Use Postulate 10 to write an equation
Subtract 5 from each side.
Divide each side by 3.
GUIDED PRACTICE for Example 1
1. Is there enough information in the diagram to conclude that m n? Explain.
ANSWER
Yes. m n because the angle corresponding to the angle measuring 75o also measures 75o since it forms a linear pair with the 105o angle. So, corresponding angles are congruent and Postulate 10 says the lines are parallel.
GUIDED PRACTICE for Example 1
2. Explain why Postulate 10 is the converse of Postulate 11.
ANSWER
Postulate 10 switches the hypothesis and conclusion of Postulate 11.
EXAMPLE 2 Solve a real-world problem
Snake Patterns
How can you tell whether the sides of the pattern are parallel in the photo of a diamond-back snake?
SOLUTION
Because the alternate interior angles are congruent, you know that the sides of the pattern are parallel.
GUIDED PRACTICE for Example 2
Can you prove that lines a and b are parallel? Explain why or why not.
3.
Yes; Alternate Exterior Angles Converse.
ANSWER
GUIDED PRACTICE for Example 2
Yes; Corresponding Angles Converse.
ANSWER
4.
GUIDED PRACTICE for Example 2
No; Supplementary angles do not have to be congruent.
ANSWER
5. m 1 + m 2 = 180°
EXAMPLE 3 Prove the Alternate Interior Angles Converse
SOLUTION
GIVEN :∠ 4 ∠5
PROVE : g h
Prove that if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
EXAMPLE 3 Prove the Alternate Interior Angles Converse
1. Given
g h4.
1. 4 ∠5
2. 1 ∠4
3. 1 ∠5
2. Vertical Angles Congruence Theorem
3. Transitive Property of Congruence
4. Corresponding Angles Converse
STATEMENTS REASONS
EXAMPLE 4 Write a paragraph proof
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.
In the figure, r s and 1 is congruent to 3. Prove p q.
EXAMPLE 4 Write a paragraph proof
Plan for Proof
a. Look at 1 and 2. Look at 2 and 3.b.
1 2 because r s. If 2 3 then p q.
EXAMPLE 4 Write a paragraph proof
Plan in Action
a. It is given that r s, so by the Corresponding Angles Postulate, 1 2.
b. It is also given that 1 3. Then 2 3 by the Transitive Property of Congruence for angles. Therefore, by the Alternate Interior Angles Converse, p q.
EXAMPLE 5 Use 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.
U.S. Flag
EXAMPLE 5 Use the Transitive Property of Parallel Lines
SOLUTION
The stripes from top to bottom can be named s1, s2, s3, . . . , s13. Each stripe is parallel to the one below it, so s1 s2, s2 s3, and so on. Then s1 s3 by the Transitive Property of Parallel Lines. Similarly, because s3 s4, it follows that s1 s4. By continuing this reasoning, s1 s13. So, the top stripe is parallel to the bottom stripe.
GUIDED PRACTICE for Examples 3, 4, and 5
6. If you use the diagram at the right to prove the Alternate Exterior Angles Converse, what GIVEN and PROVE statements would you use?
PROVE : j k
GIVEN : ∠1 8
SOLUTION
GUIDED PRACTICE for Examples 3, 4, and 5
7. Copy and complete the following paragraph proof of the Alternate Interior Angles Converse using the diagram in Example 3.
ANSWER
It is given that 4 5. By the Vertical Angle Congruence, 1 4. Then by the Transitive Property of Congruence, 1 5.So, by the Corresponding Angle Converse Postulate , g h.
It is given that 4 5. By the ? , 1 4. Then by the Transitive Property of Congruence, ? .So, by the ? , g h.
GUIDED PRACTICE for Examples 3, 4, and 5
8. 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.
ANSWER
All of the steps are parallel. Since the bottom step is parallel to the ground, the Transitive Property of Parallel Lines applies, and the top step is parallel to the ground.
