5.1
Angles and
Degree
Measures
Definitions
• An angle is formed by rotating one of two
rays that share a fixed endpoint know as the
vertex.
• The initial side of the angle is the ray that is
fixed.
• The terminal side of the angle is the ray
that rotates.
Terminal Side
Vertex
Initial Side
The measure of an
angle tells:
1. The direction
of the rotation.
2. The amount of
rotation to move
from the initial
side to the
terminal side.
• Counter-clockwise
rotation makes a
positive angle
• Clockwise rotation
makes a negative
angle.
An angle with its vertex at the
origin and its initial side along the
positive x axis is said to be in
standard position.
Angles are measured in degrees and radi
You MUST be
familiar with
measures
between 0°
and 360°
Positive angles
Negative Angles
Examples:
•
Draw an angle of 150°
in standard position.
•
Draw an angle of 225°
in standard position.
•
Draw an angle of -30°
in standard position.
•
Draw an angle of 300° in standard
position.
A full rotation around a circle is
360°. Measures more than 360°
represent multiple rotations.
2 “rotations”
Give the angle measure
represented by each rotation.
• 5.5 rotations
clockwise
-1980°
• 3.3 rotations
counterclockwise
1188°
• 9.5 rotations
clockwise
-3420°
• 6.75 rotations
counterclockwise.
2430°
A quadrantal angle is one where
the terminal side is one of the axes.
Two angles in standard position are
called coterminal angles if they have
the same terminal side.
Every angle has infinitely many
coterminal angles.
These can be found by adding or
subtracting multiples of 360°
Find one positive and one negative
angle that is coterminal with:
• 45°
• 135°
• 225°
• 300°
Identify ALL angles that are
coterminal with each angle
• 30°
• 270°
• 120°
• 330°
Determine a coterminal angle that is
between 0° and 360°. State the
quadrant in which the terminal side
lies.
• 775°
55°; QI
• 595°
235°; QIII
• -1297°
• -777°
143°; QII
303°; QIV
A reference angle is an ACUTE
angle formed by the terminal side
of the given angle and the x axis.
Find the reference angle.
• 120°
60°
• -135°
45°
• 210°
30°
• 300°
60°
Exit Pass
1. Find one positive and one negative
coterminal angle with 218°.
2. Find the reference angle for 218°
3. Draw a 300° angle in standard position.
4. Give 2 quadrantal angles.
Degrees, Minutes, Seconds
• The most common unit used to measure
angles is the degree (°).
• The degree is subdivided into 60 equal parts
known as minutes (‘).
• The minute is subdivided into 60 equal parts
known as seconds (“).
•
The concept of degree measurement
comes from the Babylonian culture.
Their number system was based off of
60 instead of 10 like we do today
Changing from Degrees to
Degrees, Minutes, and Seconds
(DMS)
• Change 15.735° to degrees, minutes, and
seconds.
15° + (0.735  60)’
15° + 44.1’
15° + 44’ + (0.1  60)”
15°+44’ + 6”
15°44’6”
Change from Degrees to DMS
• -16.75°
-16° 45’
• 27.465°
27° 27’ 54”
Change from DMS to Degrees
• Write 39° 5’ 34” as a decimal rounded to
the nearest thousdandth.
39° + 5(1/60) + 34 (1/3600)
39.093°
Change from DMS to Degrees
• 23° 14’ 30”
23.242°
• 223°25’15”
233.421°