Chapter 3Chapter 3

Perpendicular and Parallel LinesPerpendicular and Parallel Lines

3.1 – Lines and Angles3.1 – Lines and Angles

Two lines are Two lines are PARALLEL LINESPARALLEL LINES if they if they are coplanar and they do not intersectare coplanar and they do not intersect..

Two lines are Two lines are SKEW LINESSKEW LINES if they are not if they are not coplanar and they do not intersect.coplanar and they do not intersect.

Two planes that do not intersect are called Two planes that do not intersect are called PARALLEL PLANESPARALLEL PLANES..

Parallel PostulatesParallel Postulates

If there is a line and a point not on the line, If there is a line and a point not on the line, then there is exactly one line through the then there is exactly one line through the point parallel to the given line.point parallel to the given line.

P

Transversals and AnglesTransversals and Angles

A A TRANSVERSALTRANSVERSAL is a line that intersects is a line that intersects two or more coplanar lines at different two or more coplanar lines at different points.points.

Corresponding AnglesCorresponding Angles

Two angles are corresponding angles if Two angles are corresponding angles if they occupy corresponding positions.they occupy corresponding positions.

1 2

4

6

3

87

5

Alternate Interior AnglesAlternate Interior Angles

Two angles are alternate interior angles if Two angles are alternate interior angles if they lie between the two lines on opposite they lie between the two lines on opposite sides of the transversal.sides of the transversal.

1 2

4

6

3

87

5

Alternate Exterior AnglesAlternate Exterior Angles

Two angles are alternate exterior angles if Two angles are alternate exterior angles if they lie outside the two lines on opposite they lie outside the two lines on opposite sides of the transversal.sides of the transversal.

1 2

4

6

3

87

5

Consecutive Interior AnglesConsecutive Interior Angles

Two angles are consecutive int. angles if Two angles are consecutive int. angles if they lie between the two lines on the same they lie between the two lines on the same side of the transversal. (aka: same side side of the transversal. (aka: same side interior angles)interior angles)

1 2

4

6

3

87

5

Parallel, skew, or perpendicularParallel, skew, or perpendicular

UT and WT are: _________UT and WT are: _________

RS and VW are: ___________RS and VW are: ___________

TW and WX are: ___________TW and WX are: ___________

R

U

S

T

V W

X

Lines and PlanesLines and Planes

Name a line parallel to HJ.Name a line parallel to HJ.

Name a line skew to GH.Name a line skew to GH.

Name a line perpendicular to JH.Name a line perpendicular to JH.

LK

G

M

J

N

H

AnglesAngles

6 and 10 are __________ angles.6 and 10 are __________ angles. 12 and 6 are __________ angles.12 and 6 are __________ angles. 7 and 9 are __________ angles.7 and 9 are __________ angles.

67

85

9 10

1112

3.2 Perpendicular Lines3.2 Perpendicular Lines

If two lines intersect to form a linear pair of If two lines intersect to form a linear pair of congruent angles, then the lines are congruent angles, then the lines are perpendicular. perpendicular.

g h

g

h

Perpendicular LinesPerpendicular Lines

If two sides of two adjacent angles are If two sides of two adjacent angles are perpendicular, then the angles are perpendicular, then the angles are complementary.complementary.

Perpendicular LinesPerpendicular Lines

If two lines are perpendicular, then they If two lines are perpendicular, then they intersect to form four right angles.intersect to form four right angles.

3.3 Parallel Lines and Transversals3.3 Parallel Lines and Transversals

Corresponding Angles PostulateCorresponding Angles Postulate If 2 parallel lines are cut by a transversal, then If 2 parallel lines are cut by a transversal, then

the pairs of corresponding angles are the pairs of corresponding angles are congruent.congruent.

1

2

Alternate Interior AnglesAlternate Interior Angles

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

3

46

5

Consecutive Interior AnglesConsecutive Interior Angles

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

3

64

5

Alternate Exterior AnglesAlternate Exterior Angles

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

3

6 4

5

Perpendicular TransversalPerpendicular Transversal

If a transversal is perpendicular to one of If a transversal is perpendicular to one of two parallel lines, then it is perpendicular two parallel lines, then it is perpendicular to the other.to the other.

ExamplesExamples

Find m 1 and m 2.Find m 1 and m 2.

1 2

105

ExamplesExamples

Find m 1 and m 2.Find m 1 and m 2.

1 2

135

ExamplesExamples

Find x. Find x.

64 2x

ExamplesExamples

Find x.Find x.

(3x – 30)

3.4 Parallel Lines3.4 Parallel Lines

3.4 is basically the converse of 3.3.3.4 is basically the converse of 3.3.

Corresponding Angles Corresponding Angles ConverseConverse

If two lines are cut by a transversal so that If two lines are cut by a transversal so that corresponding angles are congruent, then the corresponding angles are congruent, then the lines are parallel. lines are parallel.

1

2

Alternate Interior Angles Alternate Interior Angles ConverseConverse

If two parallel lines are cut by a transversal If two parallel lines are cut by a transversal so that alternate interior angles are so that alternate interior angles are congruent, then the lines are parallel.congruent, then the lines are parallel.

34

Consecutive Interior Angles Consecutive Interior Angles ConverseConverse

If two parallel lines are cut by a transversal If two parallel lines are cut by a transversal so that consecutive interior angles are so that consecutive interior angles are supplementary, then the lines are parallel.supplementary, then the lines are parallel.

6

5

Alternate Exterior Angles Alternate Exterior Angles ConverseConverse

6

5

If two parallel lines are cut by a transversal If two parallel lines are cut by a transversal so that alternate exterior angles are so that alternate exterior angles are congruent, then the lines are parallel.congruent, then the lines are parallel.

3.5 Properties of Parallel Lines3.5 Properties of Parallel Lines

If two lines are parallel to the same line, If two lines are parallel to the same line, then they are parallel to each other.then they are parallel to each other.

p q r

Properties of Parallel LinesProperties of Parallel Lines

In a plane, if two lines are perpendicular to In a plane, if two lines are perpendicular to the same line, then they are parallel to the same line, then they are parallel to each other.each other.

m n

p

Proving Lines are Parallel

State which pair of lines are parallel and why.

m A = m 6

A

5

DB

C

F

E

6

Proving Lines are Parallel

m 6 = m 8

B

5

A C

D

8

4 6

7

Proving Lines are Parallel

1 = 90 , 5 = 90

5421

X W

ZY

Which lines are parallel if given:

1 = 4 6 = 4 2 + 3 = 5 1 = 7 1 = 8 2 + 3 + 8 = 180

l m t

j

k54

68

12

37

~

~

~

~

~

Find x and y.

Given: AB II EF , AE II BF

(3x-34)

(4y+1)

(2x-7)

D E F

CBA

3.6-3.7 Coordinate Geometry3.6-3.7 Coordinate Geometry

Slope of a line:Slope of a line:

The equation of a line:The equation of a line: Slope-intercept form: Slope-intercept form: y = mx + by = mx + b Point-slope form: Point-slope form: y – yy – y11 = m (x – x = m (x – x11))

12

12

xx

yym

Parallel and Perpendicular Parallel and Perpendicular SlopesSlopes

In a coordinate plane, two nonvertical lines In a coordinate plane, two nonvertical lines are parallel iff they have the same slope. are parallel iff they have the same slope. (Any two vertical lines are parallel.)(Any two vertical lines are parallel.)

In a coordinate plane, two nonvertical lines In a coordinate plane, two nonvertical lines are perpendicular iff the product of their are perpendicular iff the product of their slopes is -1. (Vertical and horizontal lines slopes is -1. (Vertical and horizontal lines are perpendicular.) are perpendicular.)

Parallel and Perpendicular Parallel and Perpendicular SlopesSlopes

Parallel lines have the same slopes.Parallel lines have the same slopes.

Perpendicular lines have slopes that are Perpendicular lines have slopes that are OPPOSITE RECIPROCALS. (Line 1 has OPPOSITE RECIPROCALS. (Line 1 has a slope of -2. Line 2 is perpendicular to a slope of -2. Line 2 is perpendicular to Line 1 and has a slope of ½ . Line 1 and has a slope of ½ .

Horizontal LinesHorizontal Lines

HHooririzzontal lines have a slope = ontal lines have a slope = 0 0 ((zzero)ero) The equation of a horizontal line is The equation of a horizontal line is y = by = b

Vertical LinesVertical Lines

Vertical lines have undefined slope (Vertical lines have undefined slope (undund)) The equation of a vertical line is The equation of a vertical line is x = ax = a

Finding the equation of a lineFinding the equation of a line

Write the equation of the line that goes Write the equation of the line that goes through (1,1) with a slope of 2.through (1,1) with a slope of 2.

Finding the equation of a lineFinding the equation of a line

Find the slope of the line between the Find the slope of the line between the points (0,6) and (5,2). Then write the points (0,6) and (5,2). Then write the equation of the line.equation of the line.

Parallel?Parallel?

Line 1 passes through (0,6) and (2,0).Line 1 passes through (0,6) and (2,0). Line 2 passes through (-2,6) and (0,1).Line 2 passes through (-2,6) and (0,1). Line 3 passes through (-6,5) and (-4,0)Line 3 passes through (-6,5) and (-4,0)

Perpendicular?Perpendicular?

Line 1 passes through (4,2) and (1,-4).Line 1 passes through (4,2) and (1,-4). Line 2 passes through (-1,2) and (5,-1).Line 2 passes through (-1,2) and (5,-1).

3.6-3.7 Writing equations

Write the equation of the line which passes through point (4,9) and has a slope of -2.

Equations of lines

Write the equation of the line parallel to the line y = -2/5 x + 3 and passing through the point (-5,0).

Equations of lines

Write the equation of the line parallel to the line 3x + 2y = 4 and passing through the point (-4, 5).

Equations of lines

Write the equation of the line parallel to the x-axis and passing through the point (3,-6).

Equations of lines

Write the equation of the line parallel to the y-axis and passing through the point (4,8).

Parallel Lines?

Line p1 passes through (0,-3) & (1,-2). Line p2 passes through (5,4) & (-4,-4). Line p3 passes through (-6,-1) & (3,7). Find the slope of each line. Which lines are parallel?

Slope

Find the slope of the line passing through the given points.

Parallel?

Determine if the two lines are parallel.

Slope

What is the slope of the line?

Slope

Find the slope of the line that passes through the given points.

Parallel?

Determine if the two lines are parallel.

Perpendicular?

With the given slopes, are the lines perpendicular?

m1 = 2 ; m2 = - ½

m1 = -1 ; m2 = 1

m1 = 5/7 ; m2 = - 7/5

Perpendicular?

Find the slopes of the two lines. Determine if they are perpendicular.

Perpendicular?

Line 1: y = 3x ; Line 2: y = (-1/3)x – 2

Line 1: y = (1/3)x – 10 ; Line 2: y = 3x

Perpendicular?

Line 1: 3y + 2x = -36 ; Line 2: 4y – 3x = 16

Line 1: 3y – 4x = 3 ; Line 2: 4y + 3x = -12

Writing equations

Write the equation of the line perpendicular to the line y = (-3/4)x + 6 that goes through the point (8,0)

Writing Equations

Write the equation of the line that goes through point (-3,-4) and that is perpendicular to y = 3x + 5.

Perpendicular, Parallel or Neither

y = -2x – 1 y = -2x – 3

y = 4x + 10 y = -2x + 5