Chapter 3

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# Chapter 3 - PowerPoint PPT Presentation

Chapter 3. Perpendicular and Parallel Lines. 3.1 – Lines and Angles. Two lines are PARALLEL LINES if they are coplanar and they do not intersect . Two lines are SKEW LINES if they are not coplanar and they do not intersect. Two planes that do not intersect are called PARALLEL PLANES.

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### Chapter 3

Perpendicular and Parallel Lines

3.1 – Lines and Angles
• Two lines are PARALLEL LINESif they are coplanar and they do not intersect.
• Two lines are SKEW LINES if they are not coplanar and they do not intersect.
• Two planes that do not intersect are called PARALLEL PLANES.
Parallel Postulates
• If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

P

Transversals and Angles
• A TRANSVERSAL is a line that intersects two or more coplanar lines at different points.
Corresponding Angles
• Two angles are corresponding angles if they occupy corresponding positions.

1

2

3

4

5

6

8

7

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

1

2

3

4

5

6

8

7

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

1

2

3

4

5

6

8

7

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

1

2

3

4

5

6

8

7

Parallel, skew, or perpendicular
• UT and WT are: _________
• RS and VW are: ___________
• TW and WX are: ___________

R

S

U

T

X

V

W

Lines and Planes
• Name a line parallel to HJ.
• Name a line skew to GH.
• Name a line perpendicular to JH.

M

L

K

J

N

G

H

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

6

5

7

8

10

9

11

12

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

g

h

g h

Perpendicular Lines
• If two sides of two adjacent angles are perpendicular, then the angles are complementary.
Perpendicular Lines
• If two lines are perpendicular, then they intersect to form four right angles.
3.3 Parallel Lines and Transversals
• Corresponding Angles Postulate
• If 2 parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

1

2

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

5

3

4

6

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

5

3

6

4

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

5

3

4

6

Perpendicular Transversal
• If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Examples
• Find m 1 and m 2.

1

2

105

Examples
• Find m 1 and m 2.

1

2

135

Examples
• Find x.

64

2x

Examples
• Find x.

(3x – 30)

3.4 Parallel Lines

3.4 is basically the converse of 3.3.

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

1

2

Alternate Interior Angles Converse
• If two parallel lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

3

4

Consecutive Interior Angles Converse
• If two parallel lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

5

6

Alternate Exterior Angles Converse
• If two parallel lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.

5

6

3.5 Properties of Parallel Lines
• If two lines are parallel to the same line, then they are parallel to each other.

p

q

r

Properties of Parallel Lines
• In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

p

m

n

Proving Lines are Parallel
• State which pair of lines are parallel and why.
• m A = m 6

B

D

F

5

6

A

C

E

Proving Lines are Parallel
• m 6 = m 8

B

D

8

7

5

4

6

A

C

Proving Lines are Parallel
• 1 = 90 , 5 = 90

Y

Z

1

2

4

5

X

W

1 = 4

6 = 4

2 + 3 = 5

1 = 7

1 = 8

2 + 3 + 8 = 180

Which lines are parallel if given:

~

~

l

m

t

~

j

~

8

6

~

2

1

4

5

3

k

7

Find x and y.
• Given: AB II EF , AE II BF

A

B

C

(2x-7)

(3x-34)

(4y+1)

D

F

E

3.6-3.7 Coordinate Geometry
• Slope of a line:
• The equation of a line:
• Slope-intercept form: y = mx + b
• Point-slope form: y – y1 = m (x – x1)
Parallel and Perpendicular Slopes
• In a coordinate plane, two nonvertical lines are parallel iff they have the same slope. (Any two vertical lines are parallel.)
• In a coordinate plane, two nonvertical lines are perpendicular iff the product of their slopes is -1. (Vertical and horizontal lines are perpendicular.)
Parallel and Perpendicular Slopes
• Parallel lines have the same slopes.
• Perpendicular lines have slopes that are OPPOSITE RECIPROCALS. (Line 1 has a slope of -2. Line 2 is perpendicular to Line 1 and has a slope of ½ .
Horizontal Lines
• Horizontal lines have a slope = 0 (zero)
• The equation of a horizontal line is y = b
Vertical Lines
• Vertical lines have undefined slope (und)
• The equation of a vertical line is x = a
Finding the equation of a line
• Write the equation of the line that goes through (1,1) with a slope of 2.
Finding the equation of a line
• Find the slope of the line between the points (0,6) and (5,2). Then write the equation of the line.
Parallel?
• Line 1 passes through (0,6) and (2,0).
• Line 2 passes through (-2,6) and (0,1).
• Line 3 passes through (-6,5) and (-4,0)
Perpendicular?
• Line 1 passes through (4,2) and (1,-4).
• 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.
y = -2x – 1

y = -2x – 3

y = 4x + 10

y = -2x + 5

Perpendicular, Parallel or Neither