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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

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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