Warm-Up Up Exercises. 1. Find the value of x. ANSWER 32
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1 Warm-Up Up Exercises 1. Find the value of x. ANSWER Write the converse of the following statement. If it is raining, then Josh needs an umbrella. ANSWER If Josh needs an umbrella, then it is raining.
2 3.3 Prove Lines are Parallel Warm-Up Exercises Goal: Use angle relationships to prove that lines are parallel. Key Vocabulary: paragraph proof converse two-column proof
3 3.3 Warm-Up Prove Exercises Lines are Parallel Goal: Use angle relationships to prove that lines are parallel. Postulates, Corollaries, and Theorems: Postulate 16: Corresponding Angles Converse Theorem 3.4: Alternate Interior Angles Converse Theorem 3.5: Alternate Exterior Angles Converse Theorem 3.6: Consecutive Interior Angles Converse Theorem 3.7: Transitive Property of Parallel Lines
4 Definitions Warm-Up Exercises PARAGRAPH PROOF: A type of proof written in paragraph form. The statements and reasons in a paragraph proof are written in sentences, using words to explain the logical flow of the argument.
5 Definitions Warm-Up Exercises CONVERSE: The statement formed by exchanging the hypothesis and conclusion of a conditional statement. Statement: If m A = 90!, then m A is a right angle. Converse: If m A is a right, then m A = 90!.
6 Definitions Warm-Up Exercises TWO-COLUMN PROOF: A type of proof written as numbered statements and corresponding reasons that show an argument in a logical order.
7 Warm-Up Exercises POSTULATE 16: Corresponding Angles Converse If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
8 EXAMPLE Warm-Up 1 Exercises 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 + 5) o = 65 o Use Postulate 16 to write an equation 3x = 60 x = 20 Subtract 5 from each side. Divide each side by 3. The lines m and n are parallel when x = 20.
9 Warm-Up Exercises 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 75 o also measures 75 o since it forms a linear pair with the 105 o angle. So, corresponding angles are congruent and Postulate 16 says the lines are parallel.
10 Warm-Up Exercises GUIDED PRACTICE for Example 1 2. Explain why Postulate 16 is the converse of Postulate 15. ANSWER Postulate 16 switches the hypothesis and conclusion of Postulate 15.
11 Warm-Up Exercises THEOREM 3.4: Alternate Interior Angles Converse If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
12 Prove the Alternate Interior Angles Theorem Warm-Up Exercises GIVEN: 4 5 PROVE: g h g h 1. Given 2. Vertical Angles Congruence Theorem 3. Transitive Property of Congruence 4. Corresponding Angles Converse
13 Warm-Up Exercises THEOREM 3.5: Alternate Exterior Angles Converse If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.
14 Prove the Alternate Exterior Angles Theorem Warm-Up Exercises GIVEN: 2 7 PROVE: m n m n 1. Given 2. Vertical Angles Congruence Theorem 3. Transitive Property of Congruence 4. Corresponding Angles Converse
15 Warm-Up Exercises THEOREM 3.6: Consecutive Interior Angles Converse If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
16 Prove the Consecutive Interior Angles Theorem Warm-Up Exercises GIVEN: 3 and 5 are supplementary PROVE: m n 1. 3 and 5 are supplementary 2. 5 and 6 are supplementary Given 2. Linear Pair Postulate 3. Congruent Supplements Theorem 4. m n 4. Alternate Interior Angles Converse
17 EXAMPLE Warm-Up 2 Exercises 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.
18 Warm-Up Exercises GUIDED PRACTICE for Example 2 Can you prove that lines a and b are parallel? Explain why or why not. 3. ANSWER Yes; Alternate Exterior Angles Converse.
19 Warm-Up Exercises GUIDED PRACTICE for Example 2 Can you prove that lines a and b are parallel? Explain why or why not. 4. ANSWER Yes; Corresponding Angles Converse.
20 Warm-Up Exercises GUIDED PRACTICE for Example 2 Can you prove that lines a and b are parallel? Explain why or why not. 5. m 1 + m 2 = 180 ANSWER No; Supplementary angles do not have to be congruent.
21 EXAMPLE Warm-Up 4 Exercises Write a paragraph proof In the figure, 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.
22 EXAMPLE Warm-Up 4 Exercises Write a paragraph proof Plan for Proof a. Look at 1 and 2. b. Look at 2 and because r s. If 2 3 then p q.
23 EXAMPLE Warm-Up 4 Exercises 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.
24 Warm-Up Exercises THEOREM 3.7: Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other.
25 Prove the Transitive Property of Parallel Lines Warm-Up Exercises GIVEN: p q and q r PROVE: p r 1. p q and q r Given 2. Alternate Interior Angles Theorem 3. Vertical Angles Congruence Theorem Alternate Interior Angles Theorem Transitive Property of Angle Congruence 6. p r 6. Alternate Interior Angles Converse
26 EXAMPLE Warm-Up 5 Exercises Use the Transitive Property of Parallel Lines U.S. Flag 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.
27 EXAMPLE Warm-Up 5 Exercises Use the Transitive Property of Parallel Lines SOLUTION The stripes from top to bottom can be named s 1, s 2, s 3,..., s 13. Each stripe is parallel to the one 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.
28 Warm-Up Exercises 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? ANSWER GIVEN : 1 8 PROVE : j k
29 Warm-Up Exercises 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.
30 Daily Warm-Up Homework Exercises Quiz 1. Find the value of x that makes p q. ANSWER Can you prove a b? If so, what theorem would you use? ANSWER Yes; Alternate Interior Angle Converse
31 Daily Warm-Up Homework Exercises Quiz 3. Which line are parallel. ANSWER EF DG 4. In the figure, if HJ KL and KL MN, What can you conclude? What theorem justifies your conclusion? ANSWER HJ MN by Transitive Property of Parallel Lines
32 Closing Warm-Up Exercises Lines can be proved parallel by congruent corresponding angles, alternate interior angles, or alternate exterior angles. They can also be proved parallel if consecutive interior angles are supplementary. If two lines are parallel to the same line, they are parallel to each other.
33 Closing Warm-Up Exercises You prove lines parallel by showing that corresponding angles, alternate interior angles, or alternate exterior angles are congruent, or by showing that consecutive interior angles are supplementary.
34 3.3 Warm-Up Homework Exercises P157: 1-23, 28
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