Transcript Slope and Parallel Lines

Slope and Parallel Lines
Sections 4.5 & 4.6
Definitions
• A plane is a surface such that if any two
points on the surface are connected by a
line, all points of the line are also on the
surface.
• A plane has only two dimensions – length
and width – but no thickness.
Definitions
• If points, lines, segments, and so forth lie
in the same plane, we call them coplanar.
• Points, lines, segments, and so forth that
do not lie in the same plane are called
noncoplanar.
Definitions
• A transversal is a line that intersects two
coplanar lines in two distinct points.
Definitions
• In the diagram, the
region between lines d
and e is the interior of
the figure.
• In the diagram, the rest
of the plane except the
region between lines d
and e is the exterior of
the figure.
B
B
d
A
A
e
B
B
Definitions
• Alternate Interior Angles are a pair of angles
formed by two lines and a transversal. The
angles must
– both lie in the interior of the figure,
– lie on alternate sides of the transversal,
– have different vertices.
C
D
B
A
H
F
G
E
Definitions
• Alternate Exterior Angles are a pair of angles
formed by two lines and a transversal. The
angles must
– both lie in the exterior of the figure,
– lie on alternate sides of the transversal,
– have different vertices.
C
D
B
A
H
F
G
E
Definitions
• Corresponding Angles are a pair of angles
formed by two lines and a transversal.
– One angle must lie in the interior of the figure, and the
other must lie in the exterior.
– The angles must lie on the same side
C
of the transversal but have
D
different vertices.
B
A
H
F
G
E
1
2
4 3
5
6
8
7
1
3
5
4
7
8
2
6
Parallel Lines
• Parallel (║) lines are two coplanar lines
which do not intersect.
• Parallel lines have the same slope.
Slope Review 
• The slope of a nonvertical line (or segment or
ray) containing points (x1, y1) and (x2, y2) is
defined by
y y2  y1
m

x x2  x1
• Find the slope of the line containing points
(2, -1) and (7, 4)
Remember,
• Rising line – positive slope
• Falling line – negative slope
• Horizontal line – zero slope
• Vertical line – no slope (undefined slope)
Slopes of Parallel Lines
• Theorem 26: If two nonvertical lines are
parallel, then their slopes are equal.
• Theorem 27: If the slopes of two
nonvertical lines are equal, then the lines
are parallel.
Slopes of Perpendicular Lines
• Theorem 28: If two nonvertical lines are
perpendicular, then each line’s slope is the
opposite reciprocal of the other’s.
Flip the top
and bottom of
fraction and
change to
opposite sign!
• Theorem 29: If a line’s slope is the
opposite reciprocal of another line’s slope,
then the two lines are perpendicular.