Chapter 3: Parallel and Perpendicular Lines

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Chapter 3 Parallel and Perpendicular Lines

• Lesson 1 Parallel Lines and Transversals

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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

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Frayer Model
Alternate Exterior Angles
Alternate Interior Angles
Corresponding Angles
Consecutive Interior Angles
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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
• 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

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C. Name a plane parallel to plane ABG.
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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
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A. A group of nature trails is shown. Identify
the sets of lines to which line a is a
transversal.
B. A group of nature trails is shown. Identify
the sets of lines to which line b is a
transversal.
C. A group of nature trails is shown. Identify
the sets of lines to which line c is a
transversal.
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Chapter 3 Parallel and Perpendicular Lines

• Lesson 2 Angles and Parallel Lines

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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.

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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.
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A. In the figure, a b and m?18 42. Find m?22.
B. In the figure, a b and m?18 42. Find m?25.
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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.
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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.
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Chapter 3 Parallel and Perpendicular Lines

• Lesson 3 Slopes of Lines

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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

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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

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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)

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Find the slope of the line.
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Find the slope of the line.
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Find the slope of the line.
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Find the slope of the line.
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• 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!!)

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• 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

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

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• Write the equation in slope-intercept form and
  then

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

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

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Chapter 3 Parallel and Perpendicular Lines

• Lesson 5 Proving Lines Parallel

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Two lines are parallel if they are cut by a
transversal so that (see graphic organizer)

• Corresponding angles are congruent
• Alternate exterior angles are congruent
• Consecutive interior angles are supplementary
• Alternate interior angles are congruent
• They are both perpendicular to the transversal
• 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

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A. Given ?1 ? ?3, is it possible to prove that
any of the lines shown are parallel? If so, state
the postulate or theorem that justifies your
answer.

• B. Given m?1 103 and m?4 100, is it
  possible to prove that any of the lines shown are
  parallel? If so, state the postulate or theorem
  that justifies your answer.

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A. Given ?9 ? ?13, which segments are parallel?
B. Given ?2 ? ?5, which segments are parallel?
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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