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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.

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2

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8

7

Alternate Interior Angles

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

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Alternate Exterior Angles

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

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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)

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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

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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

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Perpendicular Transversal

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

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

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

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.

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