4.5/5.2 Parallel Lines

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

4.5/5.2 Parallel Lines 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.

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

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

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

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

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

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

5 7 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. If Corr. ∠s ≅ ⇒ ∥ lines

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

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

A little review 1 2 4 3 5 6 8 7 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?

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

Example 2 M S O D E Given: ∠MSO≅∠MED Prove:

Example 3 M S O D E Given: ∠ESO supp. ∠MED Prove:

Example 4 T I E M Given: ∆TIM≅∆MET Prove: ∥

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

4.5/5.2 Parallel Lines