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Chapter 3 Parallel and Perpendicular Lines

- Lesson 1 Parallel Lines and Transversals

Definitions

- Parallel lines ( )- coplanar lines that do not

intersect (arrows on lines indicate which sets

are parallel to each other) - Parallel planes- two or more planes that do not

intersect - Skew lines- lines that do not intersect but are

not parallel (are not coplanar) - Transversal- a line that intersects two or more

lines in a plane at different points

Pairs of angles formed by parallel lines and a

transversal (see graphic organizer for examples)

- Exterior angles outside the two parallel lines
- Interior angles between the two parallel lines
- Consecutive Interior angles between the two

parallel lines, on the same side of the

transversal - Consecutive Exterior angles outside the two

parallel lines, on the same side of the

transversal - Alternate Exterior angles outside the two

parallel lines, on different sides of the

transversal - Alternate Interior angles between the two

parallel lines, on different sides of the

transversal - Corresponding angles one outside the parallel

lines, one inside the parallel lines and both on

the same side of the transversal

C. Name a plane parallel to plane ABG.

Classify the relationship between each set of

angles as alternate interior, alternate exterior,

corresponding, or consecutive interior angles

A. ?2 and ?6

B. ?1 and ?7

C. ?3 and ?8

D. ?3 and ?5

A. Identify the sets of lines to which line a is

a transversal.

B. Identify the sets of lines to which line b is

a transversal.

C. Identify the sets of lines to which line c is

a transversal.

Chapter 3 Parallel and Perpendicular Lines

- Lesson 2 Angles and Parallel Lines

If two parallel lines are cut by a transversal,

then (see graphic organizer)

- the alternate interior angles are congruent
- the consecutive interior angles are supplementary
- the alternate exterior angles are congruent
- the corresponding angles are congruent
- In a plane, if a line is perpendicular to one of

the two parallel lines, then it is also

perpendicular to the other line.

A. In the figure, m?11 51. Find m?15. Tell

which postulates (or theorems) you used.

B. In the figure, m?11 51. Find m?16. Tell

which postulates (or theorems) you used.

A. In the figure, a b and m?20 142. Find

m?22.

B. In the figure, a b and m?20 142. Find

m?23.

A. ALGEBRA If m?5 2x 10, and m?7 x 15,

find x.

B. ALGEBRA If m?4 4(y 25), and m?8 4y,

find y.

- ALGEBRA If m?1 9x 6, m?2 2(5x 3), and

m?3 5y 14, find x.

B. ALGEBRA If m?1 9x 6, m?2 2(5x 3), and

m?3 5y 14, find y.

Chapter 3 Parallel and Perpendicular Lines

- Lesson 5 Proving Lines Parallel

If (see graphic organizer)

- Corresponding angles are congruent,
- Alternate exterior angles are congruent,
- Consecutive interior angles are supplementary,
- Alternate interior angles are congruent,
- Two lines are both perpendicular to the

transversal, - Then the lines are parallel.
- If given a line and a point not on the line,

there is exactly one line through that point that

is parallel to the given line

If so, state the postulate or theorem that

justifies your answer.

A. Given ?1 ? ?3, is it possible to prove that

any of the lines shown are parallel?

- B. Given m?1 103 and m?4 100, is it

possible to prove that any of the lines shown are

parallel?.

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A. Given ?9 ? ?13, which segments are parallel?

B. Given ?2 ? ?5, which segments are parallel?

Chapter 3 Parallel and Perpendicular Lines

- Lesson 3 Slopes of Lines

Slope

- The ratio of the vertical rise over the

horizontal run - Can be used to describe a rate of change
- Two non-vertical lines have the same slope if and

only if they are parallel - Two non-vertical lines are perpendicular if and

only if the product of their slopes is -1

Foldable

- Step 1 fold the paper into 3 columns/sections
- Step 2 fold the top edge down about ½ inch to

form a place for titles. Unfold the paper and

turn it vertically. - Step 3 title the top row Slope, the middle row

Slope-intercept form and the bottom row

Point-slope form

Slope

- Rise 0 zero slope (horizontal line)
- Run 0 undefined (vertical line)
- Parallel same slope
- Perpendicular one slope is the reciprocal and

opposite sign of the other - Ex find the slope of a line containing (4, 6)

and (-2, 8)

Find the slope of the line.

Find the slope of the line.

Find the slope of the line.

Find the slope of the line.

- Determine whether FG and HJ are parallel,

perpendicular, or neither for F(1, 3), G(2,

1), H(5, 0), and J(6, 3). - (DO NOT GRAPH TO FIGURE THIS OUT!!)

- Determine whether AB and CD are parallel,

perpendicular, or neither for A(2, 1), - B(4, 5), C(6, 1), and D(9, 2)

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Chapter 3 Parallel and Perpendicular Lines

- Lesson 4 Equations of Lines

Slope-intercept form y mx b

Slope and y-intercept Two ordered-pairs (one is y-intercept) Two ordered-pairs (neither is y-intercept)

m -4 y-intercept 7 (4, 1) (0, -2) (3, 3) (2, 0)

This should be your middle row on the foldable

Point-slope form

Slope and one ordered-pair Two ordered-pairs

m (7, 2) (8, -2) (-3, -1)

This should be your bottom row on the foldable

- Write an equation in slope-intercept form of the

line with slope of 6 and y-intercept of 3.

- Write the equation in slope-intercept form and

then

- Write an equation in slope-intercept form for a

line containing (4, 9) and (2, 0).

- Write an equation in point-slope form for a line

containing (3, 7) and - (1, 3).

On the back

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Chapter 3 Parallel and Perpendicular Lines

- Lesson 6 Distance Between Parallel Lines

Perpendicular Lines and Distance

- The shortest distance between a line and a point

not on the line is the length of the

perpendicular line connecting them - Equidistant the same distance- parallel lines

are equidistant because they never get any closer

or farther apart - The distance between two parallel lines is the

distance between one line and any point on the

other line - In a plane, if two lines are equidistant from a

third line, then the two lines are parallel to

each other

Steps to find the distance between parallel lines

- Change the first equation so that the slope is

now perpendicular to the given slope. (do not

change anything else) - Set the new equation equal to the second given

equation - Solve for x.
- Plug in for x in the new equation (the one with

the perpendicular slope) and solve for y. - Find the distance between the ordered pair

created with x and y and the y-intercept from the

changed equation (the one with perpendicular

slope).

- Find the distance between each pair of lines
- y 2x 1
- y 2x - 4

- Find the distance between the two parallel lines
- y x 2
- y x -