Parallel lines and Transversals

Sec 3.1 Sol:G.3a

Lesson 2-3: Pairs of Lines 3

Parallel Lines

• Parallel lines are coplanar lines that do not intersect.• Arrows are used to indicate lines are parallel.• The symbol used for parallel lines is ||.

DC

BA

In the above figure, the arrows show that line AB is parallel to line CD.With symbols we denote, .CDAB

Lesson 2-3: Pairs of Lines 4

OBLIQUE LINES

• Oblique lines are lines that intersect, but do NOT form a right angle.

• m n

Skew Lines and Parallel Planes

Definition: Skew lines are lines that are non-coplaner and do not intersect.

Ex: What lines are skew to ?

Definition: Parallel planes areplanes that do not intersect.Ex : Name a set of parallel planes.

H

E

G

DC

BA

F

AE

Example

Lesson 2-4: Angles and Parallel Lines 7

Transversal

• Definition: A line that intersects two or more lines in a plane at different points is called a transversal.

Note: Transversals intersects do not always have to be Parallel.

• When a transversal t intersects line n and m, eight angles of the following types are formed:

Exterior anglesInterior anglesConsecutive interior anglesAlternative exterior anglesAlternative interior anglesCorresponding angles

tm

n

Exterior and Interior Angles

Exterior Angles: Angles that are on the “outside” of the two “clusters”

Interior Angles: Angles that are in the “middle “ of the two “clusters”

Consecutive Interior Angles

Consecutive Interior Angles: (Same-side Interior Angles) Are on the same side of the transversal.On the inside.

Alternate Exterior and Alternate Interior Angles

Alternate Exterior Angles: Are on the opposite sides (or they alternate sides) of the transversal.Are on the outside.

Alternate Interior Angles:Are on the opposite sides of the transversal.Are on the inside.

Corresponding Angles

Corresponding Angles: Occupy the same place in the different clusters.

m

l

t1 8

210

6

4

12

3 9

11 7

5

1. Identify each pair of angles as Alt. interior, Alt. exterior, Corresponding or Consecutive interior angles.a. b.c.d.e.f.

2. Identify all pairs of vertical angles.3. Identify all linear pairs.

51 and119 and

106 and

83 and

127 and84 and

Assignment

CW: 59-60 1-12, 14, 16-23,27Hw: pg 144-145 11-29, 37-42

Angles and Parallel Lines

Section 3.2Sol: G.3a, c, f

E.Q.: Compare and contrast parallel lines with a transversal and non-parallel lines and a transversal.

Lesson 2-4: Angles and Parallel Lines 15

Angles and Parallel Lines

• If two parallel lines are cut by a transversal, then the following pairs of angles are congruent.

1. Corresponding angles

2. Alternate interior angles

3. Alternate exterior angles

• If two parallel lines are cut by a transversal, then the following pairs of angles are supplementary.

1. Consecutive interior angles

Continued…..

Lesson 2-4: Angles and Parallel Lines 16

3-1 Corresponding angles postulate

• If two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent.

2 6, 1 5, 3 7, 4 8

1 2

3 4

5 6

7 8

Lesson 2-4: Angles and Parallel Lines 17

3-2 Consecutive Interior Angles Theroem

If two parallel lines are cut by a transversal, then then each pair of consecutive interior angles is supplementary.

m3 +m5 = 180º, m4 +m6 = 180º

1 2

3 4

5 6

7 8

Lesson 2-4: Angles and Parallel Lines 18

Alternate Angles

• 3-1 Alternate Interior Angles Thereom: If two parallel lines are cut by a transversal, then each pair of alternate interior angles are Congruent.

• 3-3 Alternate Exterior Angles: If two parallel lines are cut by a transversal, then each pair of alternate exterior angles are Congruent.

3 6, 4 5

2 7, 1 81 2

3 4

5 6

7 8

3-4 Perpendicular Transversal Thereom

• In a plane, if a line is perpendicular to one or more parallel lines, then it is perpendicular to the other.

• l||m and a||b

l

m

b

a

Assignments

CW:WB pg 63-64 1-12,14,15,16Homework:Pg 153-154 7-9,12-20,22,23-24