chapter 4 parallels. parallel lines and planes section 4-1
TRANSCRIPT
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CHAPTER 4Parallels
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Parallel Lines and Planes
Section 4-1
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Parallel Lines
Two lines are parallel if and only if they are in the same plane and do not intersect.
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Parallel Planes
Planes that do not intersect.
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Skew Lines
Two lines that are not in the same plane are skew if and only if they do not intersect.
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Parallel Lines and Transversals
Section 4-2
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Transversal
In a plane, a line is a transversal if and only if it intersects two or more lines, each at a different point.
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Alternate Interior Angles
Interior angles that are on opposite sides of the transversal
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Consecutive Interior Angles
Interior angles that are on the same side of the transversal.
Also called, same-side interior angles.
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Alternate Exterior Angles
Exterior angles that are on opposite sides of the transversal.
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Theorem 4-1
If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.
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Theorem 4-2
If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary.
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Theorem 4-3
If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.
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Transversals and Corresponding Angles
Section 4-3
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Corresponding Angles
Have different verticesLie on the same side of the transversal
One angle is interior and one angle is exterior
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Postulate 4-1
If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.
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Theorem 4-4
If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other.
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Proving Lines Parallel
Section 4-4
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Postulate 4-2
In a plane, if two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are parallel.
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Theorem 4-5
In a plane, if two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines are parallel.
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Theorem 4-6
In a plane, if two lines are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel.
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Theorem 4-7
In a plane, if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the two lines are parallel.
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Theorem 4-8
In a plane, if two lines are perpendicular to the same line, then the two lines are parallel.
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Slope
Section 4-5
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SlopeThe slope m of a line containing two points with coordinates (x1, y1) and
(x2, y2) is given by the formula
m =y2 – y1
x2 – x1
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Vertical Line
The slope of a vertical line is undefined.
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Postulate 4-3
Two distinct non-vertical lines are parallel if and only if they have the same slope.
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Postulate 4-4
Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.
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Equations of Lines
Section 4-6
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Linear EquationAn equation whose graph is a straight line.
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Y-InterceptThe y-value of the point where the lines crosses the y-axis.
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Slope-Intercept Form
An equation of the line having slope m and y-intercept b is
y = mx + b.
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Examples
Name the slope and y-intercept of each line
y = 1/2x + 5 y = 3 x = -2 2x – 3y = 18
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Examples
Graph each equation 2x + y = 3 -x + 3y = 9
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Examples
Write an equation of each line
Passes through ( 8, 6) and (-3, 3)
Parallel to y = 2x – 5 and through the point (3, 7)
Perpendicular to y = 1/4x + 5 and through the point (-3, 8)