Chapter 3: Parallel and Perpendicular Lines
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Transcript Chapter 3: Parallel and Perpendicular Lines
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
A. Name all segments parallel to BC.
B. Name a segment skew to EH.
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, m11 = 51.
Find m15. Tell which
postulates (or theorems) you
used.
B. In the figure, m11 = 51.
Find m16. Tell which
postulates (or theorems) you
used.
A. In the figure, a || b and
m20 = 142. Find m22.
B. In the figure, a || b and
m20 = 142. Find m23.
A. ALGEBRA If m5 = 2x – 10,
and m7 = x + 15, find x.
B. ALGEBRA If m4 = 4(y – 25),
and m8 = 4y, find y.
A.
ALGEBRA If m1 = 9x +
6, m2 = 2(5x – 3), and
m3 = 5y + 14, find x.
B. ALGEBRA If m1 = 9x + 6,
m2 = 2(5x – 3), and
m3 = 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 m1 = 103 and
m4 = 100, is it possible to
prove that any of the lines
shown are parallel?.
Find ZYN so that
||
. Show your work.
A. Given 9 13, which segments are parallel?
B. Given 2 5, which segments are parallel?
___
__
C. Find x so that AB || HI if m1 = 4x + 6 and
m14 = 7x – 27.
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 “Slopeintercept form” and the bottom row “Point-slope form”
Slope
y2 y1
m
x2 x1
Rise = 0
Run = 0
zero slope (horizontal line)
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)
A. Graph the line that contains Q(5, 1) and is
parallel to MN with M(–2, 4) and N(2, 1).
B. Graph the line that contains (-1, -3) and is
perpendicular to MN for M(–3, 4) and N(5, –8)?
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: y y1 m( x x1 )
Slope and one ordered-pair
m = 1
(7, 2)
Two ordered-pairs
(8, -2)
3
* This should be your bottom row on the foldable
(-3, -1)
Write an equation in slope-intercept
form of the line with slope of 6 and yintercept of –3.
Write the equation in slope-intercept form and then
Write an equation in point-slope form of the line
whose slope is
graph the line.
that contains (–10, 8). 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:
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:
1.
2.
3.
4.
5.
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=
y=
1
4
1
4
x+2
9
x4