Download Geometry - Chapter 03 Summary

Geometry
Chapter 3
Key Vocabulary
Parallel Lines – coplanar lines that do not intersect
Skew Lines – non-coplanar lines that do not intersect
Parallel Planes – planes that do not intersect
Transversal – line that intersects two or more coplanar lines at different points
Angles Formed by Transversals
Corresponding Angles (CA)
1 2
3 4
Alternate Exterior Angles (AEA)
Alternate Interior Angles (AIA)
6 5
7 8
Consecutive Interior Angles (CIA)
x intercept – where graph intersects the x axis; x -intercept (a,0)
y intercept – where graph intersects the y axis; y -intercept (0, b)
Postulates/ Theorems
Postulate 13 – Parallel Postulate
If there is a line and a point not on the line, then there exists only one line through the point parallel to
the given line.
Postulate 14 – Perpendicular Postulate
If there is a line and a point not on the line, then there exists only one line through the point
perpendicular to the given line.
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Geometry
Chapter 3
Postulate 15 – Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the corresponding
angles are congruent.
If p q , then
1 5
Theorem 3.1 – Alternate Interior Angles Theorem
1 2
4 3
If two parallel lines are cut by a transversal, then the alternate
interior angles are congruent.
If p q , then
p
4 6
5 6
Theorem 3.2 – Alternate Exterior Angles Theorem
8
q
7
If two parallel lines are cut by a transversal, then the alternate
exterior angles are congruent.
If p q , then
2 8
Theorem 3.3 – Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then the consecutive
interior angles are supplementary.
If p q , then m 3  m 6  180
Postulate 16 – Corresponding Angles Converse
If two lines are cut by a transversal so that corresponding angles are
congruent, then the lines are parallel.
If
1  5 , then a b
Theorem 3.4 – Alternate Interior Angles Converse
If two lines are cut by a transversal so that alternate interior angles are
congruent, then the lines are parallel.
If
3  6 , then a b
a
b
1 2
3 4
5 6
8 7
Theorem 3.5 – Alternate Exterior Angles Converse
If two lines are cut by a transversal so that alternate exterior angles are
congruent, then the lines are parallel.
If
2  8 , then a b
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Geometry
Chapter 3
Theorem 3.6 – Consecutive Interior Angles Converse
If two lines are cut by a transversal so that consecutive interior angles
are supplementary, then the lines are parallel.
If m 3  m 5  180 , then a b
Theorem 3.7 – Transitive Property of Parallel Lines
p
q
r
If two lines are parallel to the same line, then they are
parallel to each other.
If p q and q r , then p r
Postulate 17 – Slopes of Parallel Lines


Parallel lines have the same slope
Vertical lines are parallel
Postulate 18 – Slopes of Perpendicular Lines



Slopes of perpendicular lines are opposite reciprocals
Vertical and horizontal lines are perpendicular
Product of slopes is -1
Theorem 3.8
e
If two lines intersect to form a linear pair of congruent angles, then the lines
are perpendicular.
If
1  2 , then e  f
1 2
Theorem 3.9
f
a
If two lines are perpendicular, then they intersect to form four right angles.
If a  b , then it creates for right angles
b
Theorem 3.10
If two sides of two adjacent acute angles are perpendicular, then the angles
are complementary.
If j k , then
1, 2 are complementary
j
1
2
k
3
Geometry
Chapter 3
Theorem 3.11 – Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the other.
If h k and j  h , then j  k
Theorem 3.12 – Lines Perpendicular to a Transversal Theorem
m
n
In a plane, it two lines are perpendicular to the same line,
then they are parallel to each other.
p
If m  p and n  p , then m n
Concepts
Slopes of Lines
Positive Slope – rises from left to right
Negative Slope – falls from left to right
Slope of Zero – horizontal line
Undefined Slope – vertical line
Slope-Intercept Form of a Line
y  mx  b
Standard Form of a Line
Ax  By  C
Formulas
Slope Formula – used to find slope of line given two points
Slope 
rise
run
or
m
y2  y1
x2  x1
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