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1
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Definition
of Angle |
An
angle is the union of two rays
that have the same endpoint.
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2
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The
Angle Measure Postulate |
- Unique
Measure Assumption:
Every angle has a unique measure from zero degrees to 180 degrees.
- Unique
Angle Assumption:
- Zero
Angle Assumption:
- Straight
Angle Assumption:
- Angle
Addition Property:
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3
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Definition
of Angle Bisector |
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4
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Definition
of Central Angle of a Circle |
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5
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Definitions
of Measure of Minor and Major Arcs |
*Note:
The degree measure of an arc indicates an amount of turn and is
different than arc length which is an actual distance, or
portion of the circumference of a circle.
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6 |
Rotations: |
Rotations are turns.
They are measured with arc measure, and the direction of a rotation
is determined by its sign.
Clockwise rotations
are always negative (-).
Counterclockwise rotations are always positive (+).
*Note: The magnitude
of a rotation can be negative, but the magnitude of an angle (by
the Angle Measurement Postulate) can't be negative.
Rotations also can have
magnitudes outside the range of -180 to 180 degrees, such as 400
degrees. This can always be converted to a rotation in the given
range by adding or subtracting a multiple of 360 degrees. A rotation
of 400 degrees will land the user in the same spot as one of 40
degrees.
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|
Definitions
of Types of Angles |
If m is the
measure of an angle, then the angle is:
- zero
if and only if m=0
- acute
if and only if 0 < m < 90
- right
if
and only if m = 90
- obtuse
if and only if 90 < m < 180
- straight
if and only if m = 180
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8
|
Definitions
of Complementary and Supplementary Angles |
If the measures
of two angles are s and t, then the angles are:
- complementary
if and only if s +t = 90
- supplementary
if and only if s + t =180
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9
|
Definition
of Adjacent Angles |
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10
|
Definition
of Linear Pair |
Two
adjacent angles form a linear pair
if and only if their non-common sides are opposite rays.
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11
|
Linear
Pair Theorem |
If
two angles form a linear pair, then they are supplementary. |
12
|
Definition
of Vertical Angles |
Two
non-straight angles are vertical angles
if and only if the union of their sides is two lines.
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Angles
4 and 2 are vertical, and so are angles 1 and 3. |
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13
|
Vertical Angles Theorem |
If
two angles are vertical angles, then they have equal measures. |
14
|
Postulates of Equality |
For any real
numbers a, b, and c:
- Reflexive
Property of Equality:
a = a.
- Symmetric
Property of Equality:
If a = b, then b = a.
- Transitive
Property of Equality:
If a= b and b = c, then a = c.
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15
|
Equation
to Inequality Property |
If a and b are positive,
real, numbers and a + b = c, then c > a and c
> b. |
16
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Corresponding
Angles Postulate |
Suppose two
coplanar lines are cut by a transversal.
- If two corresponding
angles have the same measure, then the lines are parallel.
- If the lines
are parallel, then corresponding angles have the same measure.
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17
|
Definition
of Slope |
When a linear
equation is written in "slope intercept
form" it is solved for y and looks like this: y
= mx + b. The number "m", in front of the x,
is the slope, or steepness of the line. The number in the "b"
spot is the y intercept, the place where the line crosses the y
axis.
HORIZONTAL
LINES have zero slope.
VERTICAL LINES have undefined
slope.
LINES WITH NEGATIVE SLOPE appear to be going "downhill"
as you look at them from left to right.
LINES WITH POSITIVE SLOPE appear to be going "uphill"
as you look at them from left to right.
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18
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Parallel
Lines and Slopes Theorem |
Two
nonvertical lines are parallel if and only if they have the same slope. |
19
|
Transitivity
of Parallelism Theorem |
In
a plane, if line s is parallel to line m and line m
is parallel to line t, then line s is parallel to line
t. |
20
|
Definition
of Perpendicular |
Two
segments, rays, or lines are perpendicular
if and only if the lines containing them form a 90 degree angle. |
21
|
Two
Perpendiculars Theorem |
If
two coplanar lines a and b are each perpendicular to
the same line, then they are parallel to each other.
|
22
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Perpendicular
to Parallels Theorem |
In
a plane, if a line is perpendicular to one of two parallel lines,
then it is also perpendicular to the other. |
23
|
Perpendicular
Lines and Slopes Theorem |
Two
nonvertical lines are perpendicular if and only if the product of
their slopes is -1. |
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