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3.3 Prove Lines are Parallel Goal Your Notes p Use angle relationships to prove that lines are parallel. VOCABULARY Paragraph proof POSTULATE 16 CORRESPONDING ANGLES CONVERSE If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are . Example 1 2 j 6 k Apply the Corresponding Angles Converse Find the value of x that makes m i n. 718 Solution Lines m and n are parallel if the marked corresponding angles are congruent. (2x 1 3)8 5 (2x 1 3)8 m n Use Postulate 16 to write an equation. 2x 5 Subtract x5 from each side. Divide each side by The lines m and n are parallel when x 5 . . Checkpoint Find the value of x that makes a i b. 1. (5x 2 7)8 988 Copyright © Holt McDougal. All rights reserved. a b Lesson 3.3 •Geometry Notetaking Guide 69 3.3 Prove Lines are Parallel Goal Your Notes p Use angle relationships to prove that lines are parallel. VOCABULARY Paragraph proof A proof can be written in paragraph form, called a paragraph proof. POSTULATE 16 CORRESPONDING ANGLES CONVERSE If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel . Example 1 2 j 6 k Apply the Corresponding Angles Converse Find the value of x that makes m i n. Solution Lines m and n are parallel if the marked corresponding angles are congruent. (2x 1 3)8 5 718 2x 5 68 718 (2x 1 3)8 m n Use Postulate 16 to write an equation. Subtract 3 from each side. x 5 34 Divide each side by 2 . The lines m and n are parallel when x 5 34 . Checkpoint Find the value of x that makes a i b. x 5 21 1. (5x 2 7)8 988 Copyright © Holt McDougal. All rights reserved. a b Lesson 3.3 •Geometry Notetaking Guide 69 Your Notes 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 . j 4 5 k 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 . 1 j k 8 THEOREM 3.6 CONSECUTIVE INTERIOR ANGLES CONVERSE If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are . Example 2 j 3 5 k Solve a real-world problem Flags How can you tell whether the sides of the flag of Nepal are parallel? Solution Because the are congruent, you know that the sides of . the flag are Checkpoint Can you prove that lines a and b are parallel? Explain why or why not. 2. m∠1 1 m∠2 5 1808 1 2 a b 70 Lesson 3.3 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes 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 . j 4 5 k 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 . 1 j k 8 THEOREM 3.6 CONSECUTIVE INTERIOR ANGLES CONVERSE If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel . Example 2 j 3 5 k Solve a real-world problem Flags How can you tell whether the sides of the flag of Nepal are parallel? Solution Because the alternate interior angles are congruent, you know that the sides of the flag are parallel . Checkpoint Can you prove that lines a and b are parallel? Explain why or why not. 2. m∠1 1 m∠2 5 1808 1 2 a b Yes, you can use the Consecutive Interior Angles Converse to prove a i b. 70 Lesson 3.3 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Write a paragraph proof Example 3 In the figure, a i b and ∠1 is congruent to ∠3. Prove x i y. a b 2 1 Solution 3 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. a 3 a 1 b 2 x y 1 3 because a i b. In paragraph proofs, transitional words such as so, then, and therefore help to make the logic clear. y b. Look at ∠2 and ∠3. b 2 x x y If ∠2 > ∠3 then . Plan in Action a. It is given that a i b, so by the , ∠1 > ∠2. by the b. It is also given that ∠1 > ∠3. Then Transitive Property of Congruence for angles. Therefore, , x i y. by the Checkpoint Complete the following exercise. 3. In Example 3, suppose it is given that ∠1 > ∠3 and x i y. Complete the following paragraph proof showing that a i b. It is given that x i y. By the Exterior Angles . Postulate, It is also given that ∠1 > ∠3. Then by the Transitive Property of Congruence for angles. Therefore, by the , a i b. Copyright © Holt McDougal. All rights reserved. Lesson 3.3 • Geometry Notetaking Guide 71 Your Notes Write a paragraph proof Example 3 In the figure, a i b and ∠1 is congruent to ∠3. Prove x i y. a b 2 1 Solution 3 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. a 3 a 1 b 2 x y 1 3 ∠1 > ∠2 because a i b. In paragraph proofs, transitional words such as so, then, and therefore help to make the logic clear. y b. Look at ∠2 and ∠3. b 2 x x y If ∠2 > ∠3 then x iy . Plan in Action a. It is given that a i b, 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 Exterior Angles Converse , x i y. Checkpoint Complete the following exercise. 3. In Example 3, suppose it is given that ∠1 > ∠3 and x i y. Complete the following paragraph proof showing that a i b. It is given that x i y. By the Exterior Angles Postulate, ∠2 > ∠3 . It is also given that ∠1 > ∠3. Then ∠1 > ∠2 by the Transitive Property of Congruence for angles. Therefore, by the Corresponding Angles Converse , a i b. Copyright © Holt McDougal. All rights reserved. Lesson 3.3 • Geometry Notetaking Guide 71 Your Notes THEOREM 3.7 TRANSITIVE PROPERTY OF PARALLEL LINES p q r If two lines are parallel to the same line, then they are to each other. Example 4 Use the Transitive Property of Parallel Lines Utility poles Each utility pole shown is parallel to the pole immediately to its right. Explain why the leftmost pole is parallel to the rightmost pole. When you name several similar items, you can use one variable with subscripts to keep track of the items. t1 t2 t3 t4 t5 t6 Solution The poles from left to right can be named t1, t2, t3, . . . , t6. , Each pole is parallel to the one to its right, so t1 i t2 i , and so on. Then t1 i t3 by the . Similarly, because t3 i t4, it . By continuing this reasoning, t1 i . follows that t1 i So, the leftmost pole is parallel to the rightmost pole. Checkpoint Complete the following exercise. 4. Each horizontal piece of the window blinds shown is called a slat. Each slat is parallel to the slat immediately below it. Explain why the top slat is parallel to the bottom slat. s1 s16 Homework 72 Lesson 3.3 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes THEOREM 3.7 TRANSITIVE PROPERTY OF PARALLEL LINES p q r If two lines are parallel to the same line, then they are parallel to each other. Example 4 Use the Transitive Property of Parallel Lines Utility poles Each utility pole shown is parallel to the pole immediately to its right. Explain why the leftmost pole is parallel to the rightmost pole. When you name several similar items, you can use one variable with subscripts to keep track of the items. t1 t2 t3 t4 t5 t6 Solution The poles from left to right can be named t1, t2, t3, . . . , t6. Each pole is parallel to the one to its right, so t1 i t2 , t2 i t3 , and so on. Then t1 i t3 by the Transitive Property of Parallel Lines . Similarly, because t3 i t4, it follows that t1 i t4 . By continuing this reasoning, t1 i t6 . So, the leftmost pole is parallel to the rightmost pole. Checkpoint Complete the following exercise. 4. Each horizontal piece of the window blinds shown is called a slat. Each slat is parallel to the slat immediately below it. Explain why the top slat is parallel to the bottom slat. Homework 72 s1 s16 The slats from top to bottom can be named s1, s2, s3, . . . , s16. Each slat is parallel to the one below it, so s1 i s2, s2 i s3, and so on. Then s1 i s3 by the Transitive Property of Parallel Lines. Similarly, because s3 i s4, it follows that s1 i s4. By continuing this reasoning, s1 i s16. So, the top slat is parallel to the bottom slat. Lesson 3.3 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.