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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

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Chapter 3

Chapter 3

Perpendicular and Parallel Lines


3 1 lines and angles

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

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

Transversals and Angles

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


Corresponding angles

Corresponding Angles

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

1

2

3

4

5

6

8

7


Alternate interior angles

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

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

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

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

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

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

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

Perpendicular Lines

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


Perpendicular lines1

Perpendicular Lines

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


3 3 parallel lines and transversals

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 angles1

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 angles1

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 angles1

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

Perpendicular Transversal

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


Examples

Examples

  • Find m 1 and m 2.

1

2

105


Examples1

Examples

  • Find m 1 and m 2.

1

2

135


Examples2

Examples

  • Find x.

64

2x


Examples3

Examples

  • Find x.

(3x – 30)


3 4 parallel lines

3.4 Parallel Lines

3.4 is basically the converse of 3.3.


Corresponding angles converse

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

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

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

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

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

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

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 parallel1

Proving Lines are Parallel

  • m 6 = m 8

B

D

8

7

5

4

6

A

C


Proving lines are parallel2

Proving Lines are Parallel

  • 1 = 90 , 5 = 90

Y

Z

1

2

4

5

X

W


Which lines are parallel if given

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

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

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

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 slopes1

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

  • Horizontal lines have a slope = 0 (zero)

  • The equation of a horizontal line is y = b


Vertical lines

Vertical Lines

  • Vertical lines have undefined slope (und)

  • The equation of a vertical line is x = a


Finding the equation of a line

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 line1

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

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

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

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

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 lines1

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 lines2

Equations of lines

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


Equations of lines3

Equations of lines

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


Parallel lines

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

Slope

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


Parallel1

Parallel?

  • Determine if the two lines are parallel.


Slope1

Slope

  • What is the slope of the line?


Slope2

Slope

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


Parallel2

Parallel?

  • Determine if the two lines are parallel.


Perpendicular1

Perpendicular?

  • With the given slopes, are the lines perpendicular?

  • m1 = 2 ; m2 = - ½

  • m1 = -1 ; m2 = 1

  • m1 = 5/7 ; m2 = - 7/5


Perpendicular2

Perpendicular?

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


Perpendicular3

Perpendicular?

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

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


Perpendicular4

Perpendicular?

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

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


Writing equations

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 equations1

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

Perpendicular, Parallel or Neither


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