Angles and Their Measures
Chapter 4, Sections 1 & 3
Angles
 An angle is formed by two rays that have a
common endpoint called the vertex.
 One ray is called the initial side and the other
the terminal side.
 The arrow near the vertex shows the direction
and the amount of rotation from the initial side to
the terminal side.
C
A
è
Initial Side
Terminal Side
B
Vertex
Standard Position
An angle is in standard position if
 its vertex is at the origin of a rectangular
coordinate system and
 its initial side lies along the positive x-axis.
Standard Position – Positive Angles
y
 is positive
Terminal Side

x
Vertex
Initial Side
Positive angles rotate counterclockwise.
Standard Position – Negative Angles
y
Vertex
Initial Side
x
Terminal Side

 is negative
Negative angles rotate clockwise.
Measuring Angles
 We measure angles in 2 different units: degrees
and radians
 Degrees
Divided into 60 equal parts called minutes (‘)
 Divided into 60 equal parts called seconds (“)
 Radians
Uses π
Angle Conversions – Degrees to DMS
1. Keep the whole number – this is the
degree part (ᵒ)
2. Multiply the decimal part by 60 – this is
the minutes part (‘)
3. Multiply the remaining decimal part by 60
– this is the seconds part (“)
Angle Conversions – DMS to Degrees
1. Keep the degree part – this is the whole
number.
2. Divide the minutes part by 60, divide the
seconds part by 3600, then add those two
numbers to each other – this is the
decimal.
Angle Conversions – Degrees to Radians
1. Divide the degrees by 180
2. Multiply by π
**leave in fraction form**
**if the angle is in DMS form, convert back
to degrees first**
Angle Conversions – Radians to Degrees
Divide by π
Multiply by 180
Special Angles – Coterminal Angles
two angles that share a terminal side
To find coterminal angles – add or subtract
any number of full circles to the angle
The degree measure of an angle has been
increased/decreased by a multiple of 360º
The radian measure of an angle has been
increased/decreased by a multiple of 2π
Special Angles – Reference Angles
the acute angle (A) formed by the terminal
side of the given angle and the x-axis
In Quadrant I, the reference angle is A
In Quadrant II, the reference angle is 180-A
In Quadrant III, the reference angle is A-180
In Quadrant IV, the reference angle is 360-A
Special Angles – Reference Angles
Special Angles – Quadrantal Angles
the terminal side of the angle coincides
with one of the axes
90º
180º
270º
360º
In Conclusion
Exit Slip – Summarize what you’ve learned
about angles and their measures using a
bubble map.
Homework – Page 358
Problems 2-22 even