1. 4.5/5.2 Parallel Lines I can prove lines parallel I can recognize planes and transversals I can identify the pairs of angles that are congruent given parallel lines cut by a transversal Day 2 Find x, y, and z. Find the perimeter of PBRE

2. What is a plane? ► Defn: A plane is a flat surface that continues infinitely in all directions. A plane has no height, only a length and a width. 4.5/5.2 Parallel Lines

3. Types of lines ► There are two types of lines associated with planes. 1.Coplanar – lines, segments, or rays that lie in the same plane 2.Noncoplanar – lines, segments, or rays that do not lie in the same plane ► Lines that are coplanar can either be intersecting or parallel

4. ► Identify the transversal in the following diagram a b c ► Defn: A transversal is a line that intersects two coplanar lines.

5. The regions of intersecting lines ► The region between two lines is called the interior ► Everything else is the exterior a b interior exterior

6. Parallel Lines ► Parallel line are two coplanar lines that never intersect. ► The two lines MUST be coplanar.

7. Alternate interior angles ► ► Alternate interior angles are two angles in the interior of a figure on opposite sides of the transversal. 1 2 a b c If Alt. int. ∠ s ≅ ⇒ ∥ lines

8. Alternate exterior angles ► ► Alternate exterior angles are angles that lie in the exterior of a figure on the opposite sides of the transversal. If Alt. ext. ∠ s ≅ ⇒ ∥ lines 1 8 c a b

9. Corresponding angles ► ► Corresponding angles are angles on the same side of the transversal where one angle is in the interior and one in the exterior. 5 7 If Corr. ∠ s ≅ ⇒ ∥ lines

10. Let’s see what you know ► Based on the following diagram, name all pairs of… 1. Alternate interior angles 2. Alternate exterior angles 3. Corresponding angles 4. Vertical angles 1 2 3 4 5 6 7 8 a b c

11. 6 ways to prove lines parallel 2. Alt. ext. ∠ s ≅⇒∥ 4. Same side int. ∠ s supp. ⇒∥ 5. Same side ext. ∠ s supp. ⇒∥ 6. 2 lines ⊥ to same line ⇒∥ 3. Corr. ∠ s ≅⇒∥ 1. Alt. int. ∠ s ≅ ⇒ ∥ lines 1 2 3 4 5 6 7 8 a b c

12. ► Which angle is alt. int. with ∠ 3? ► Which angle is alt. ext. with ∠ 1? ► Which angle is corresponding with ∠ 4? ► Which angle is same side int. with ∠ 5? ► Which angle is same side ext. with ∠ 1? A little review 8 7 65 43 21

13. ► State the theorem used to prove m ∥ n. Example 1 5 1 ∠ 1 ≅∠ 5 m n 5 4 ∠ 4 supp. ∠ 5 m n 7 1 ∠ 1 ≅∠ 7 m n

14. Example 2 M S E D O

15. Example 3 M S E D O

16. Example 4 T EM I

17. ► Solve for x. Justify that a ∥ b. Example 5 (10x-14) ∘ (8x+2) ∘ (5x+26) ∘ a b