slope and parallel lines sections 4.5 & 4.6. definitions a plane is a surface such that if any...
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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.
e
d
BB
BB
AA
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.
F
BA
D
HE
C
G
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.
F
BA
D
HE
C
G
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 of the transversal but have different vertices.
F
BA
D
HE
C
G
8 7
65
4 3
21
87
65
4
3
2
1
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
• Find the slope of the line containing points (2, -1) and (7, 4)
2 1
2 1
y yym
x x x
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.
• Theorem 29: If a line’s slope is the opposite reciprocal of another line’s slope, then the two lines are perpendicular.
Flip the top and bottom of fraction and change to
opposite sign!