2-9 Proving Lines Parallel

1 Given the following information, determine which Lines , if any, are Parallel . State the postulate or theorem that justifies your answer. 1. SOLUTION: and are corresponding angles of Lines j and , j || k by the Converse of Corresponding Angles Postulate. ANSWER: j || k; converse of corresponding angles postulate 2. SOLUTION: and are alternate interior angles of Lines j andk. Since , j || k by the Converse of Alternate Interior Angles Theorem. ANSWER: j || k; alternate interior angles converse 3. SOLUTION: and are alternate exterior angles of Lines and m. Since , || m by the Converse of Alternate Exterior Angles Theorem. ANSWER: alternate exterior angles converse 4.

2 M6 + m8 = 180 SOLUTION: and are consecutive interior angles of Lines and m. Since , by the Converse of Consecutive Interior Angles Theorem. ANSWER: consecutive interior angles converse 5. Find x so that m || n. Identify the postulate or theorem you used. SOLUTION: angle and angle are alternate exterior angles of Lines m and n. Since m || n, by the Converse of Alternate Exterior Angles Theorem. Solve for x. ANSWER: 20, Converse of Alternate Exterior Angles Theorem 6. PROOF Copy and complete the proof of Theorem Given: Prove: Proof: SOLUTION: ANSWER: 7. RECREATION Is it possible to prove that the backrest and footrest of the lounging beach chair are Parallel ?

3 If so, explain how. If not, explain why not. SOLUTION: Yes it it possible to prove that the backrest and footrest of the lounging beach chair are Parallel . Since the alternate exterior angles are congruent, the backrest and footrest are Parallel . ANSWER: Sample answer: Yes; since the alternate exterior angles are congruent, the backrest and footrest are Parallel . Given the following information, determine which Lines , if any, are Parallel . State the postulate or theorem that justifies your answer. 8. SOLUTION: and are corresponding angles of Lines r and , r || s by the Converse of Corresponding Angles Postulate. ANSWER: r || s; Converse of Corresponding Angles Postulate 9.

4 SOLUTION: and are alternate exterior angles of Lines u and v. Since , u || v by the Converse of Alternate Exterior Angles Theorem. ANSWER: u || v; Alternate Exterior Angles Converse 10. SOLUTION: and are alternate interior angles of Lines r ands. Since , r || s by the Converse of Alternate Interior Angles Theorem. ANSWER: r || s; Alternate Interior Angles Converse 11. SOLUTION: and are consecutive interior angles of Lines r and s. Since , r || s by the Converse of Consecutive Interior AnglesTheorem. ANSWER: r || s; Consecutive Interior Angles Converse 12. SOLUTION: and are consecutive interior angles of Lines u and v. Since , u || v by the Converse of Consecutive Interior AnglesTheorem.

5 ANSWER: u || v; Consecutive Interior Angles Converse 13. SOLUTION: and are alternate interior angles of Lines u and v. Since , u || v by the Converse of Alternate Interior Angles Theorem. ANSWER: u || v; Alternate Interior Angles Converse 14. SOLUTION: No Lines can be proven : No Lines can be proven ||. 15. SOLUTION: and are corresponding angles of Lines r and , r || s by the Converse of Corresponding Angles Postulate. ANSWER: r || s; Corresponding Angles Converse Find x so that m || n. Identify the postulate or theorem you used. 16. SOLUTION: By the Alternate Exterior Angles Converse, if 3x 14 = 2x + 25, then m || n. Solve for x. ANSWER: 39; Alt.

6 Ext. s Conv. 17. SOLUTION: By the Converse of Corresponding Angles Postulate,if 5x 20 = 90, then m || n. Solve for x. ANSWER: 22; Conv. Corr. s Post. 18. SOLUTION: By the Alternate Interior Angles Converse, if 21 + 2x= x + 84, then m || n. Solve for x. ANSWER: 63; Alt. Int. s Conv. 19. SOLUTION: By the Consecutive Interior Angles Converse, if 7x 2 + 10 3x = 180, then m || n. Solve for x. ANSWER: 43; Consec. Int. s Conv. 20. SOLUTION: Use the Vertical Angle Theorem followed by Consecutive Interior Angles Converse to find x. Then by Consecutive Interior Angles Converse, if 3x+ 2x + 45 = 180, then m || n. Solve for x. ANSWER: 27; Vert.

7 S Thm and Consec. Int. s Conv. 21. SOLUTION: By the Alternate Exterior Angles Converse, if 6x 144 = 2x, then m || n. Solve for x. ANSWER: 36; Alt. Ext. s Conv. 22. SENSE-MAKING Wooden picture frames are often constructed using a miter box or miter saw. These tools allow you to cut at an angle of a given size. If each of the four pieces of framing material is cut at a 45 angle, will the sides of the frame be Parallel ? Explain your reasoning. SOLUTION: Yes, If each of the four pieces of framing material is cut at a 45 angle, then the sides of the frame be Parallel ; when two pieces are put together, they forma 90 angle. Two Lines that are perpendicular to the same line are Parallel .

8 ANSWER: Yes; when two pieces are put together, they form a 90 angle. Two Lines that are perpendicular to the same line are Parallel . 23. PROOF Copy and complete the proof of Theorem Given: 1 and 2 are supplementary. Prove: SOLUTION: ANSWER: 24. CRAFTS Jacqui is making a stained glass piece. She cuts the top and bottom pieces at a 30 angle. If the corners are right angles, explain how Jacqui knows that each pair of opposite sides are Parallel . SOLUTION: Since the corners are right angles, each pair of opposite sides is perpendicular to the same line. Therefore, each pair of opposite sides is Parallel . ANSWER: Since the corners are right angles, each pair of opposite sides is perpendicular to the same line.

9 Therefore, each pair of opposite sides is Parallel . PROOF Write a two-column proof for each of the following. 25. Given: Prove: SOLUTION: Proof: Statements (Reasons) 1. , (Given) 2. (Corr. postulate) 3. (Trans. Prop.) 4. (If alternate are , then Lines are .) ANSWER: Proof: Statements (Reasons) 1. 1 3, (Given) 2. 2 3 (Corr. s postulate) 3. 1 2 (Trans. Prop.) 4. (If alternate s are , then Lines are .) 26. Given: Prove: SOLUTION: Proof: Statements (Reasons) 1. 2 3 (Given) 2. 2 and 4 are supplementary. (Cons. Int. s) 3. m 2 + m 4 = 180 (Def. of suppl. s) 4. m 3 + m 4 = 180 (Substitution) 5. 3 and 4 are supplementary.

10 (Def. of suppl. s) 6. (If cons. int. s are suppl., then Lines are .) ANSWER: Proof: Statements (Reasons) 1. 2 3 (Given) 2. 2 and 4 are supplementary. (Cons. Int. s) 3. m 2 + m 4 = 180 (Def. of suppl. s) 4. m 3 + m 4 = 180 (Substitution) 5. 3 and 4 are supplementary. (Def. of suppl. s) 6. (If cons. int. s are suppl., then Lines are .) 27. Given: Prove: SOLUTION: Proof: Statements (Reasons) 1. , (Given) 2. (Def. of ) 3. (Substitution) 4. and are supplementary. (Def. of suppl. ) 5. (If consec. int. are suppl., then Lines are .) ANSWER: Proof: Statements (Reasons) 1. ABC ADC, mA + mABC = 180 (Given) 2. mABC = mADC (Def.)