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Name
Class
Date
10.1 Slope and Parallel Lines
Essential Question: How can you use slope to solve problems involving parallel lines?
Resource
Locker
Explore
Proving the Slope Criteria for
Parallel Lines
The following theorem states an important connection between slope and parallel lines.
Theorem: Slope Criteria for Parallel Lines
Two nonvertical lines are parallel if and only if they have the same slope.
Follow these steps to prove the slope criteria for parallel lines.
A
First prove that if two lines are parallel, then they have the same slope.
Suppose lines m and n are parallel lines that are neither vertical nor horizontal.
Let A and B be two points on line m, as shown. You can draw a horizontal line
through A and a vertical line through B to create the “slope triangle,”△ABC.
_
You can extend AC to intersect line n at point D and then extend it to point F
so that AC = DF. Finally, you can draw a vertical line through F intersecting
line n at point E.
Mark the figure to show parallel lines, right angles, and congruent segments.
© Houghton Mifflin Harcourt Publishing Company
B
m
E
D
A
F
C
B
n
When parallel lines are cut by a transversal, corresponding angles are congruent, so
∠BAC ≅
∠EDF .
△BAC ≅ △EDF by the ASA Triangle Congruence Theorem.
_
¯
By CPCTC, BC ≅ EF and BC = EF .
BC
EF
The slope of line m = _, and the slope of line n = _.
DF
AC
The slopes of the lines are equal because the numerators of the fractions are equal and the
denominators of the fractions are equal (Division Property of Equality).
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CNow prove that if two lines have the same slope, then they are parallel.
Suppose lines m and n are two lines with the same nonzero slope. You can set up a
figure in the same way as before.
Let A and B be two points on line m, as shown. You can draw a horizontal line
through A and a vertical line through B to create the “slope triangle,”△ABC .
_
You can extend AC to intersect line n at point D and then extend it to point F
so that AC = DF. Finally, you can draw a vertical line through F intersecting line n
at point E.
Mark the figure to show right angles and congruent segments.
B
m
E
D
A
F
C
D
n
EF
BC
Since line m and line n have the same slope, _ = _.
DF
AC
BC
EF
But DF = AC, so by substitution, _ = _.
AC
AC
Multiplying both sides by AC shows that BC = EF .
Now you can conclude that △BAC ≅ △EDF by the SAS Triangle Congruence
Theorem.
By CPCTC, ∠BAC ≅
∠EDF
.
Line m and line n are two lines that are cut by a transversal so that a pair of
corresponding angles are congruent.
You can conclude that line m is parallel to line n .
1.
Explain why the slope criteria can be applied to horizontal lines.
If two parallel lines are horizontal, then both lines have a slope of 0. If two lines both have
a slope of 0, then both lines are horizontal, so they are parallel.
2.
Explain why the slope criteria cannot be applied to vertical lines even though all
vertical lines are parallel.
The slope of a vertical line is undefined. So, you can’t say that two vertical lines have the
same slope.
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Reflect
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Using Slopes to Classify Quadrilaterals
by Sides
Explain 1
You can use the slope criteria for parallel lines to analyze figures in the coordinate plane.
Example 1 Show that each figure is the given type of quadrilateral.
 Show that
ABCD is a trapezoid.
4
Step 1 Find the coordinates of the vertices of quadrilateral ABCD.
A(−1, 1), B(2, 3), C(3, 1), D(−3, −3)
A
_
Step 2 Use the slope
formula
to
find
the
slope
of
AB
and the
_
slope of DC.
-4
D
_ y2 - y1
3- 1
2
_
_
slope of AB = _
x 2 - x 1 = 2 - (-1) = 3
y
B
2
0
-2
C
2
x
4
-4
_ y2 - y1
1 - (-3)
4
2
_
_
_
slope of DC = _
x 2 - x 1 = 3 - (-3) = 6 = 3
Step 3 Compare the slopes.
_
_
Since the slopes are the same, AB is parallel to DC.
Quadrilateral
ABCD is a trapezoid because it is a quadrilateral with at
least one pair of parallel sides.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©BuildPix/
Construction Photography/Alamy
 Show that
P
PQRS is a parallelogram.
4
2
y
Q
x
-4
-2 S 0
-2
2
4
R
-4
Step 1 Find the coordinates of the vertices of quadrilateral PQRS.
(3
P(-3, 4), Q(1, 2), R
)(
)
-2 , S -1 0
,
Step 2 Use the slope formula to find the slope of each side.
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,
_ _
y2 - y1
1
2- 4 = _
-2 = -_
PQ: x - x = _
2
1
4
2
(
)
1 - -3
-2 - 2
-4
_ _
y2 - y1
QR: x - x = _ = _ = -2
2
1
3 -1
2
2
0 - -2
1
_ _
y2 - y1
RS: x - x =_
=_=-_
2
1
2
-4
-1 - 3
4
4- 0
_ _
y2 - y1
SP: x - x = _ = _ = -2
2
1
3 - -1
-2
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Step 3 Compare the slopes.
_
Since the slope of PQ is the same as the slope of
_
Since the slope of QR is the same as the slope of
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_
¯
¯
RS , PQ
RS .
is parallel to
_
¯
¯
SP , QR
is parallel to SP .
Quadrilateral PQRS is a parallelogram because both pairs of opposite sides are parallel .
\ Reflect
3.
_
_
What If? _
Suppose you know that the lengths of PQ and QR in the figure in Example 1B
are each √20 . What type of parallelogram is quadrilateral PQRS? Explain.
PQRS is a rhombus; Since ¯
PQ ≅ ¯
QR and opposite sides of a parallelogram are congruent,
you can conclude that all four sides are congruent. A quadrilateral with four congruent
sides is a rhombus.
Your Turn
Show that each figure is the given type of quadrilateral.
4.
Show that JKLM is a trapezoid.
J(−2, 3), K(4, 1), L(−1, 0), M(-4, 1)
1- 3
1
-2 = -_
______
= ___
3
6
4 - (-2)
1
1- 0
1
________
___
¯
slope of LM =
=
=- _
2
slope of ¯
JK =
-4 - (-1)
-3
y
4
J
K
x
M
-4
3
slopes are the same, so ¯
JK is parallel to ¯
LM
-2 L 0
-2
2
4
-4
Quadrilateral JKLM is a trapezoid because it is a
quadrilateral with at least one pair of parallel sides.
5.
Show that ABCD is a parallelogram.
4
_ 4 - (-2)
3 ; slope of _
2 - 4 = -2
BC = _
slope of AB = _ = _
2
-1
2
1- (-3)
_ -4 - 2
-4 - (-2)
3 ; slope of _
slope of CD = _
DA = __ = -2
=_
2
-2 - 2
-2 - (-3)
Quadrilateral ABCD is a parallelogram because both pairs of
opposite sides are parallel.
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C
x
-4
0
2
4
A
D
-4
© Houghton Mifflin Harcourt Publishing Company
A(−3, 2), B(1, 4), C(2, 2), D(-2, 4)
yB
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Explain 2
Using Slopes to Find Missing Vertices
Example 2 Find the coordinates of the missing vertex in each parallelogram.

△ABCD with vertices A(1, −2), B(−2, 3),and D(5, −1)
y
B 4
Step 1 Graph the given points.
_
Step 2 Find the slope of AB by counting units from A to B.
2
x
The rise from -2 to 3 is 5. The run from 1 to -2 is -3.
-2
0
2
-2
Step 3 Start at D and count the same number of units.
-3
-3
_ _
Step 4 Use the slope formula to verify that BC|| AD.
_
4- 3 = _
1
slope of BC = _
4
2 - (-2)
_ -1 - (-2)
1
slope of AD = _ = _
4
5- 1
The coordinates of vertex
C are (2, 4).
5
5
-2
0
-2
6
C
B 4
Label (2, 4) as vertexC.
D
A
y
A rise of 5 from −1 is 4. A run of −3 from 5 is 2.
4
2
4
D
x
6
A
 ▱PQRS with vertices P(−3, 0), Q(−2, 4),and R(2, 2)
Step 1 Graph the given points.
_
Step 2 Find the slope of PQ by counting units from Q to P.
© Houghton Mifflin Harcourt Publishing Company
The rise from 4 to 0 is -4 . The run from
−2 to −3 is -1 .
Q
Step 3 Start at R and count the same number of units.
-1 from 2 is 1 .
A rise of -4 from 2 is -2 . A run of
Label
( 1 , -2 ) as vertex S.
P
-4 -1
-2
_ _
Step 4 Use the slope formula to verify that QR|| PS.
2 - 4
1
_
_
_
slope of
=slope of QR =
2 - -2
The coordinates of vertex
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S are
(1
,
-2
)
2
4
y
2
-4
0
-2
R
-4
S
2
x
4
-1
-4
-2 - 0
1
_
_
_
PS =
=1 - -3
2
.
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Reflect
6.
― ―
Discussion In Part A, you used the slope formula to verify that BC ∥ AD. Describe another
way you can check that you found the correct coordinates of vertex C. _ _
Possible answer: You could use the slope formula to verify thatAB ǁ CD . The slope of
_
_
5
5
AB is -__
and the slope of CD is -__
, so the sides are parallel.
3
3
Your Turn
Find the coordinates of the missing vertex in each parallelogram.
7.
▱JKLM with vertices J(−3, −2), K(0, 1),
8.
and M(1, −3)
4
-2
J
4
E
-3
Lx
0 3
3
-2
-4
and G(3, −2)
y
2
K
-4
▱DEFG with vertices E(−2, 2), F(4, 1),
2
M
4
-4
3
3
y
2
-2 0
D -1
-2
F
x
2
-4
4
-3
G -1
From F to G, rise = -3; run = -1; D is at (-3, -1).
_
1-2
1
= -__
;
Check: slope of EF = ________
6
4 -(-2)
_ -2 -(-1)
1
slope of DG = _________ = -__
6
From J to K, rise = 3; run = 3; L is at (4, 0).
_ 0-1
1
Check: slope of KL = _____
= -__
;
4
4-0
_
3 -(-2)
1
slope of JM = -______ = -__
4
1 - (-3)
3 -(-3)
Elaborate
9.
quadrilateral is a trapezoid. Once you have found that pair, you can stop finding slopes.
10. A student was asked to determine whether quadrilateral ABCD with vertices A(0, 0),
B(2, 0), C(5, 7), and D(0, 2) was a parallelogram. Without plotting points, the student
looked at the coordinates of the vertices and quickly determined that quadrilateral
ABCD could not be _
a parallelogram. How do you think the student solved the problem?
Possible answer: AD is horizontal, so in order for ABCD to be a parallelogram, the slope of
_
_
the opposite side, BC, would have to be 0. However, the slope of BC is 7 - 0 = 7 ≠ 0.
_____ __
5-2
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Suppose you are given the coordinates of the vertices of a quadrilateral. Do you always
need to find the slopes of all four sides of the quadrilateral in order to determine
whether the quadrilateral is a trapezoid? Explain.
No; you only need to find one pair of opposite parallel sides to conclude that the
3
11. Essential Question Check-In What steps can you use to determine whether two given
lines on a coordinate plane are parallel?
Possible answer: Find two points on each line and determine the slope of each line. If the
slopes are equal, then the lines are parallel.
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