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Chapter 3: Parallel and Perpendicular Lines

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Chapter 3: Parallel and Perpendicular Lines Lesson 1: Parallel Lines and Transversals – PowerPoint PPT presentation

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Title: Chapter 3: Parallel and Perpendicular Lines


1
Chapter 3 Parallel and Perpendicular Lines
  • Lesson 1 Parallel Lines and Transversals

2
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

3
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

4
C. Name a plane parallel to plane ABG.
5
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
6
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.
7
Chapter 3 Parallel and Perpendicular Lines
  • Lesson 2 Angles and Parallel Lines

8
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.

9
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.
10
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.
11
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.
12
  1. 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.
13
Chapter 3 Parallel and Perpendicular Lines
  • Lesson 5 Proving Lines Parallel

14
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

15
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?.

16
(No Transcript)
17
A. Given ?9 ? ?13, which segments are parallel?
B. Given ?2 ? ?5, which segments are parallel?
18
Chapter 3 Parallel and Perpendicular Lines
  • Lesson 3 Slopes of Lines

19
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

20
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

21
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)

22
Find the slope of the line.
23
Find the slope of the line.
24
Find the slope of the line.
25
Find the slope of the line.
26
  • 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!!)

27
  • Determine whether AB and CD are parallel,
    perpendicular, or neither for A(2, 1),
  • B(4, 5), C(6, 1), and D(9, 2)

28
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29
Chapter 3 Parallel and Perpendicular Lines
  • Lesson 4 Equations of Lines

30
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
31
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
32
  • Write an equation in slope-intercept form of the
    line with slope of 6 and y-intercept of 3.

33
  • Write the equation in slope-intercept form and
    then

34
  • Write an equation in slope-intercept form for a
    line containing (4, 9) and (2, 0).

35
  • Write an equation in point-slope form for a line
    containing (3, 7) and
  • (1, 3).

36
On the back
37
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38
Chapter 3 Parallel and Perpendicular Lines
  • Lesson 6 Distance Between Parallel Lines

39
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

40
Steps to find the distance between parallel lines
  1. Change the first equation so that the slope is
    now perpendicular to the given slope. (do not
    change anything else)
  2. Set the new equation equal to the second given
    equation
  3. Solve for x.
  4. Plug in for x in the new equation (the one with
    the perpendicular slope) and solve for y.
  5. 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).

41
  • Find the distance between each pair of lines
  • y 2x 1
  • y 2x - 4

42
  • Find the distance between the two parallel lines
  • y x 2
  • y x -
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