.
m.
.
Theorem 9.10bGivenLinesl1and l2with slope ProveProofChoose any point on
m
is
k
.
EXAMPLE 1The vertices of quadrilateral ABCDare A(2,4),B(6,2),C(2, 6), and D(1, 2).a.Show that two sides of the quadrilateral are parallel.
b.Show that the quadrilateral has two right angles.
21ml1l2If the slopes of two non-vertical lines in the coordinate plane are equal, thenthe lines are parallel.
Parallel Lines in the Coordinate Plane343
l1kl2xy

Solution
The slope of
.
The slope of
.
The slope of
.
The slope of
.
a.
and
are parallel because they have
equal slopes.
b.
The slope of
is the negative reciprocal of
the slope of
, so they are perpendicular.
Therefore,
B
is a right angle.
The slope of
is the negative reciprocal of the slope of
, so they are
perpendicular. Therefore,
A
is a right angle.
Answers
a.
b.
A
and
B
are right angles.
EXAMPLE 2
Write an equation for
l
1
, the line through (
2, 5) that is parallel to the line
l
2
whose equation is 2
x
y
7.
Solution
(1) Solve the equation of
l
2
for
y
:
(2)
Find the slope of
l
2
. The slope of a line in
slope-intercept form is the coefficient of
x
:
(3)
Find the slope of
l
1
, which is equal to the
slope of
l
2
:
(4)
Use the definition of slope to write an equation
of
l
1
. Let (
x, y
) and (
2, 5) be two points
on
l
1
:
Answer
y
2
x
1 or 2
x
y
1
DA
BC
DA
AB
BC
AB
DA
BC
DA
5
2
4
2
2
2
2
(
2
1
)
5
2
6
3
5 2
2
CD
5
2
2
6
2
1
2
2
5
2
4
2
3
5
4
3
BC
5
6
2
(
2
2
)
2
2
6
5
8
2
4
5 2
2
AB
5
2
2
2
(
2
4
)
6
2
2
5
2
4
5
1
2
344
Parallel Lines
O
C
B
A
D
1
y
x
1
2
x
y
7
y
2
x
7
y
2
x
7
slope of
l
1
2
slope
2
y
5
2(
x
2)
y
5
–
2
x
4
y
2
x
1
y
2
5
x
2
(
2
2
)
y
1
2
y
2
x
1
2
x
2
slope

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