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GEOMETRY
UNIT 1: ANGLES
ANGLES AND POINTS
An Angle is a figure formed by two rays with a common endpoint,
called the vertex.
ray
vertex
ray
Angles can have points in the interior, in the exterior or on the
angle.
A
E
D
B
C
Example: Points A, B and C are on the angle. D is in the interior
and E is in the exterior. B is the vertex.
3 WAYS TO NAME AN ANGLE
We can name an angle using:
(1) Using 3 points
(2) Using 1 point
(3) Using a number
Naming an Angle
Using 3 points: vertex must be the middle letter
This angle can be named as ÐABC or ÐCBA
Using 1 point: using only vertex letter
* Use this method is permitted when the vertex point is the vertex
of one and only one angle.
Since B is the vertex of only this angle, this can
also be called
.
ÐB
B
A
C
NAMING AN ANGLE CONTINUED
Using a number:
A
B
2
C
A number (without a degree symbol) may be
used as the label or name of the angle. This
number is placed in the interior of the angle
near its vertex. The angle to the left can be
named as Ð2 .
* The “1 letter” name is unacceptable when …
more than one angle has the same vertex point. In this case, use
the three letter name or a number if it is present.
EXAMPLE

K is the vertex of more than one angle.
Therefore, there is NOÐK in this diagram.
There is ÐLKM , ÐPKM , and ÐLKP
There is also Ð2 and Ð3 but there is no Ð5!!!
L
M
2
K
3
P
4 TYPES OF ANGLES
Acute Angle: an angle whose measure is less than 90°.
Right Angle: an angle whose measure is exactly
90°.
Obtuse Angle: an angle whose measure is between
90° and 180°.
Straight Angle: an angle that is exactly 180°.
MEASURING ANGLES
Just as we can measure segments, we can also measure angles.
We use units called degrees to measure angles.
360º
•
A circle measures _____
•
A (semi) half-circle measures _____
•
90º
A quarter-circle measures _____
•
One degree is the angle measure of 1/360th of a circle.
ADDING ANGLES
When you want to add angles, use the notation m1, meaning
the measure of 1.
If you add m1 + m2, what is your result?
m1 + m2 = 58°.
m1 + m2 = mADC also.
Therefore, mADC = 58°.
ANGLE ADDITION POSTULATE
Postulate: The sum of the two smaller angles will always equal
the measure of the larger angle.
Complete:
M
K
m  MRK
____ + m KRW
____ = m  MRW
_____
W
R
EXAMPLE: ANGLE ADDITION
K is interior to MRW, m  MRK = (3x), m KRW = (x + 6)
and mMRW = 90º. Find mMRK.
First, draw it!
K
M
W
3x
x+6
R
3x + x + 6 = 90
4x + 6 = 90
– 6 = –6
4x = 84
x = 21
Are we done?
mMRK = 3x = 3•21 = 63º
ANGLE BISECTOR
An angle bisector is a ray in the interior of an angle that
splits the angle into two congruent angles.
Example: Since 4   6, UK is an angle bisector.
5
3
CONGRUENT ANGLES
Definition: If two angles have the same measure, then they are
congruent.
Congruent angles are marked with the same number of “arcs”.
The symbol for congruence is .
Example:
3   5.
3
5
Example…
Draw your own diagram and answer this question:
If ML is the angle bisector of PMY and
mPML = 27°, then find:
mPMY = _______
mLMY = _______