Parallel Line Properties

1. PARALLEL LINE PROPERTIES

2. WORKING TOGETHER Draw two parallel lines using lined paper or the two edges of a ruler. Then draw a transversal that intersects the two parallel lines obliquely. Label all the angles from 1 to 8. List all pairs of corresponding, alternate interior, alternate exterior , same side interior and same side exterior angles.

3. WORKING TOGETHER Observe the measurements and make conjectures about the measures of the angle pairs. Write your conclusions in if-then form. Compare your results with those of your classmates.

4. If parallel lines have a transversal, then… ( l ║ m ) ( n ) Corresponding angles are congruent Alternate interior angles are congruent Alternate exterior angles are congruent 1 2 3 4 5 6 7 8 m l n 1 5 2 6 3 7 4 8 3 5 4 6 1 7 2 8

5. If a transversal ( n ) cuts two parallel lines ( l ║ m ) , then all pairs of interior angles on the same side of the transversal are supplementary. 3 + 6 = 180 4 + 5 = 180 They are same-side interior angles. If a transversal ( n ) cuts two parallel lines ( l ║ m ) , then all pairs of exterior angles angles on the same side of the transversal are supplementary. 2 + 7 = 180 1 + 8 = 180 They are same-side exterior angles. 1 2 3 4 5 6 7 8 m l n

6. If a transversal is perpendicular to one of two parallel lines, then it is also perpendicular to the other. Given: l ║ m , l ┴ n Thus, m ┴ n 1 5 m l n ┌

7. SUMMARY Corresponding angles are congruent. PCAC Theorem Alternate interior angles are congruent. PAIC * Parallel lines are indicated as lines with the same number of arrowheads located near the center.

8. Alternate exterior angles are congruent. PAEC Theorem Interior angles on the same side of a transversal are supplementary. PSSIAS Theorem Exterior angles on the same side of a transversal are supplementary. PSSEAS Theorem 3 4 j h 3 4 1 2 1 + 2 = 180 4 3 3 + 4 = 180

9. If a transversal obliquely intersects two parallel lines, then: All the acute angles are congruent. All the obtuse angles are congruent. Any pair of acute and obtuse angle is supplementary.

10. SUMMARY If parallel lines are cut by a transversal, then Corresponding angles are congruent. Alternate interior angles are congruent. Alternate exterior angles are congruent. Same side interior angles are supplementary. Same side exterior angles are supplementary.

11. Sample Problems 1. 2. (8x + 34) (5x - 10) a b (7x + 54) (3x + 90) Same Side Interior Angles supplementary angles Alternate Interior Angles congruent angles

12. Sample Problems 3. 4. Alternate Exterior Angles congruent angles Corresponding Angles congruent angles (8x – 8) o (6x + 48) o (8x + 2) o (5x + 20) o

13. Seatwork Answer nos. 20 – 22 in your E-Math book page 112 in ½ paper (crosswise)

14. The importance of learning this… Take a look around you. Chances are, you can see an example of parallel lines from where you are sitting. But how can you be sure the lines you see are parallel? Architects and builders use the basic geometric concepts in this chapter to insure that lines are indeed parallel.

15. PROVING LINES PARALLEL p g 91

16. If two lines have… a transversal and a pair of congruent corresponding angles, a transversal and a pair of congruent alternate interior angles, a transversal and a pair of congruent alternate exterior angles, i nterior angles on the same side of the transversal that are supplementary, exterior angles on the same side of the transversal that are supplementary, If two coplanar lines are perpendicular to the same line, then the lines are parallel

17. Examples 1. 2. 5. 130 130 ┌ ┌ ┌ 70 YES, alternate interior angles are congruent NO. The corresponding angles are not congruent. Therefore, they are not parallel. YES, because the two lines are perpendicular to the same line.

18. TRANSVERSALS IN REALITY