Warm-Up

1. Warm-Up • Using the lines on a piece of paper as a guide, draw a pair of parallel lines. Now draw a transversal that intersects the parallel lines. Label the angles with numbers.

2. Warm-Up • Place a patty paper over the set of angles ∠1, ∠2, ∠3, and ∠4 and copy the two intersecting lines onto the patty paper. • Slide the patty paper down and compare angles 1 through 4 with angles 5 through 8. 2 1 4 3

3. Warm-Up • Place a patty paper over the set of angles ∠1, ∠2, ∠3, and ∠4 and copy the two intersecting lines onto the patty paper. • Slide the patty paper down and compare angles 1 through 4 with angles 5 through 8. 2 1 4 3

4. Warm-Up Do you notice a relationship between pairs of corresponding, alternate interior, and alternate exterior angles? 2 1 4 3

5. Use Parallel Lines and Transversals Objectives: • To find angle pair measurements with parallel lines cut by a transversal • To prove theorems involving parallel lines cut by a transversal

6. Parallel Lines Two lines are parallel lines if and only if they are coplanar and never intersect. The red arrows indicate that the lines are parallel.

7. Parallel Lines Two lines are parallel lines if and only if they are coplanar and never intersect.

8. Skew Lines Two lines are skew lines if and only if they are not coplanar and never intersect.

9. Example 2 Think of each segment in the figure as part of a line. Which line or plane in the figure appear to fit the description? • Line(s) parallel to CD and containing point A. • Line(s) skew to CD and containing point A.

10. Example 2 • Line(s) perpendicular to CD and containing point A. • Plane(s) parallel plane EFG and containing point A.

11. Transversal A line is a transversal if and only if it intersects two or more coplanar lines. • When a transversal cuts two coplanar lines, it creates 8 angles, pairs of which have special names

12. Transversal • ∠1 and ∠5 are corresponding angles • ∠3 and ∠6 are alternate interior angles • ∠1 and ∠8 are alternate exterior angles • ∠3 and ∠5 are consecutive interior angles

13. Example 3 Classify the pair of numbered angles.

14. Example 4 List all possible answers. • ∠2 and ___ are corresponding ∠s • ∠4 and ___ are consecutive interior ∠s • ∠4 and ___ are alternate interior ∠s

15. Four Window Foldable Corresponding Angles Postulate If two parallel lines are cut by a transversal, then pairs of corresponding angles are congruent. Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent.

16. Four Window Foldable Alternate Exterior Angle Theorem If two parallel lines are cut by a transversal, then pairs of alternate exterior angles are congruent. Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then pairs of consecutive interior angles are supplementary.

17. Example 1 On the map below, 1st and 2nd Ave. are parallel.

18. Example 1 A city planner proposes to locate a small garden on the triangular island formed by the intersections of the four streets below.

19. Example 1 What are the measures of the three angles of the garden?

20. Example 2: SAT In the figure, if l || m, what is the value of x?

21. Example 3: SAT In the figure, if l1 || l2 and l3 || l4, what is y in terms of x.

22. Example 4 Prove the Alternate Interior Angle Theorem. Given: Prove:

23. Example 5 Given: and Prove:

24. Example 6 Calculate each lettered angle measure.

25. Example 7 Find the values of x and y if k || l || m.