4.5/5.2 Parallel LinesI 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

What is a plane?►Defn: A plane is a flat surface that

continues infinitely in all directions.x

yA plane has no height, only a length and a width.

4.5/5.2 Parallel Lines

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

►Identify the transversal in the following diagram

a

b

c

► Defn: A transversal is a line that intersects two coplanar lines.

The regions of intersecting lines

► The region between two lines is called the interior

► Everything else is the exterior

14

21a

b

interior

exterior

exterior

Parallel Lines

►Parallel line are two coplanar lines that never intersect.

►The two lines MUST be coplanar.

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

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

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

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

34

56

78

a

b

c

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

34

56

78

a

b

c

►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

8 765

4 321

►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

►Given: ∠MSO≅∠MED►Prove:

Example 2

M

S

ED

O

►Given: ∠ESO supp. ∠MED►Prove:

Example 3

M

S

ED

O

►Given: ∆TIM≅∆MET►Prove: ∥

Example 4

T

E M

I

►Solve for x. Justify that a∥b.

Example 5

(10x-14)∘

(8x+2)∘

(5x+26)∘

a

b