Proving Lines Parallel 3-4C

1. Proving Lines Parallel 3-4C You found slopes of lines and used them to identify parallel and perpendicular lines. p. 207 • Recognize angle pairs that occur with parallel lines. • Prove that two lines are parallel.

2. Are these lines parallel? Is there information in this figure to assume that the lines are parallel? What information from the last section can we use to show that these lines are parallel. What do we need to add first?

3. Are these lines parallel? t a b If a transversal is added to the picture, what information would tell you these lines are parallel. (There are four different things you can pick.)

4. Angles that can be used to show parallel lines… • Alternate exterior angles • Alternate interior angles • Corresponding angles • Same side interior angles that add to 180°

5. a 104° 104° b 104° c List the pair of parallel lines. How do you know the lines are parallel? a ll b by alt. int. angles are congruent b ll c by corresponding angles are congruent a ll c by alt. int. angles are congruent

6. Converse of Corresponding Angles Postulate If two lines are cut by a transversal so that a pair of corresponding angles are congruent, then the lines are parallel. Page 207

7. Euclid’s Postulate This is one of Euclid’s five original postulates from 300 BC. Postulate 2.1 and Theorem 2.10 are similar to two of Euclid’s other postulates. Page 208

8. Page 208

9. A contractor wants to guarantee that the new street she is marking of is parallel to Douglass Street. She finds that the measure of angle 1 is 45°. Give three different ways that she can use angles to be sure that the new street is parallel to Douglass Street. Main St. Douglass St. 1 2 3 4 New St. 5

10. A. Given 1  3, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer. 1 and 3 are corresponding angles of lines a and b. Answer: Since 1  3, a║b by the Converse of the Corresponding Angles Postulate.

11. B. Given m1 = 103 and m4 = 100, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer. 1 and 4 are alternate interior angles of lines a and c. Answer: Since 1 is not congruent to 4, line a is not parallel to line c by the Converse of the Alternate Interior Angles Theorem.

12. A. Given 1  5, is it possible to prove that any of the lines shown are parallel? A. Yes; ℓ║ n B. Yes; m ║ n C. Yes; ℓ║ m D. It is not possible to prove any of the lines parallel.

13. Find mZYN so that || . Show your work. Read the Test Item From the figure, you know that mWXP = 11x – 25 and mZYN = 7x + 35. You are asked to find mZYN. Solve the Test ItemWXP and ZYN are alternate exterior angles. For line PQ to be parallel to line MN, the alternate exterior angles must be congruent. SomWXP = mZYN. Substitute the given angle measures into this equation and solve for x. Once you know the value of x, use substitution to findmZYN.

14. m WXP = m ZYN Alternate exterior angles Since mWXP = mZYN,WXP ZYNand || . 11x – 25 = 7x + 35 Substitution 4x – 25 = 35 Subtract 7x from each side. 4x = 60 Add 25 to each side. x = 15 Divide each side by 4. Now use the value of x to findmZYN. mZYN = 7x + 35 Original equation = 7(15) + 35 x= 15 = 140 Simplify. Answer:mZYN = 140

15. How can we show (prove) that lines are parallel? By showing that alternate exterior angles, alternate interior, corresponding angles or same-side interior angles are congruent.

16. 3-5 Assignment Page 211, 6, 8-20 even