Unit 20
ANGULAR MEASURE
TERMS DEFINED

An angle is a figure made by two lines that intersect
 An angle is also described as the union of two rays
having a common end point
 The two rays are called the sides of the angle, and
their common end point is called a vertex
 Angles are measured in degrees. The degree symbol
is °; the angle symbol is 
 The degree of precision required in measuring and
computing angles depends on how the angle is used
 A complete circle equals 360°
2
UNITS OF ANGULAR MEASURE


The decimal degree is generally the preferred unit of
measurement in metric calculations
In the United States it is customary to express angular
measure in the following ways:






As decimal degrees, such as 9.7 degrees
As fractional degrees, such as 36 1/2 degrees
As degrees, minutes, and seconds, such as 63 degrees, 27
minutes, 48 seconds
A degree is divided in 60 equal parts called minutes ()
A minute is divided in 60 equal parts called seconds
()
So we now have a conversion between degrees,
minutes and seconds
3
Converting to decimal degrees
EXPRESSING
DEGREES,
We can use unity fraction
method or
dimensional analysis
like in units 8 &AS
9
MINUTES,
AND SECONDS

Start with theDEGREES
seconds and convert to
DECIMAL

minutes.
 Then convert the minutes into degrees
and add the decimal onto your degrees.
4
CONVERSION EXAMPLE

Express 73°5748 as
decimal degrees:


48"
1'
Convert the 48 to
x
 0 .8 '
minutes
1
60"
Add that to your minutes




So 57.8’
Now change that to
degrees
Add to the 73°
So 73.963°
57.8' 1
x
 0.963
1
60'
5
CONVERSION EXAMPLE


Express 48.54 as degrees, minutes, and seconds (DMS):
You know you have the 48° already so you can work with the
decimal.




Convert the decimal to minutes
0.54 60'
x
 32.4'
1
1
So we have 32’
Now convert that decimal minutes to seconds
0.4' 60"
x
 24"
1
1'
So 48° 32’ 24”
6
ADDING ANGLES
•
Add the following angles:
49° 53 37
+ 38 47 24 Add seconds to seconds, minutes to
minutes, etc.
87° 100 61
–
Simplify the sum: 61 =
1 1
– Now add the 1 to the 100 or 100 + 1 = 101
– Change 101 to degrees: 101 = 1 41
– Add the 1 degree to the 87 degrees and combine all the units:
87 + 1 = 88, so we end up with 88 41 1 Ans
7
Converting on Calculators

Scientific calculators will do the tedious
work with DMS and decimal degrees.
The easiest way to find how to do it is to
refer to your manual or google DMS and
you calculator model. I am fairly familiar
with all the calculators out there so if you
have trouble please get a hold of me. I
will explain some of the commonly used
calculators in class.
8
•
Subtract the following angles:
SUBTRACTING
ANGLES
89 23 15
– 70 35 20
–
–
–
20 cannot be subtracted from 15, so borrow
60 from the 23
35 cannot be subtracted from the 22 that were
left after borrowing, so borrow 60 from the 89
Complete the subtraction 88 82 75
– 70 35 20
18 47 55 Ans
9
MULTIPLYING ANGLES
•
Multiply 51 33 42 by 3:
51 33 42
 3
153° 99 126
–
Simplify the product
126 = 2 6
– Adding these 2 to the 99, we now have 101
101 = 1 41
– Adding this degree to the 153 degrees, we now have 154°
– Combining units, we now have 154 41 6 Ans
10
DIVIDING ANGLES

Divide 147 55 34 by 2:
73 57 47 Ans
2 147 55' 34"
146
1°= 60
+ 55
115
114
1= 60
+ 34
94
94
11
Two angles are complementary
COMPLEMENTS
AND when their
sum is 90°
SUPPLEMENTS



Two angles are supplementary when their
sum is 180°
Determine the complement of 43° 18:

Subtract 43 18 from 90 (or 89 60) to
find its complement
89 60
– 43 18
46 42 Ans
12
PRACTICE PROBLEMS

Express the following degrees, minutes, and
seconds as decimal degrees. Round to three
decimal places where necessary:
1.
2.

Express the following decimal degrees as degrees,
minutes, and seconds:
3.
4.

143 54 32
242 33 24
129.76°
85.845°
Perform the following operations. Simplify all
answers:
5.
6.
45° 54 39 + 79 17 43
87 16 25 – 76 21 36
13
PRACTICE PROBLEMS (Cont)
54 47 32 + 16 19 35
8. 72 15 15 – 60 20 20
9. 43 33 29  3
10. 54 48 15  3
11. 136 58 45  4
12. 272 38 52  4
13. Determine the complement of 49 15 16
14. Determine the supplement of 49 15 16
15. Determine the supplement of 147 36 21
7.
14
PROBLEM ANSWER KEY
1.
2.
3.
4.
5.
6.
7.
8.
143.909°
242.557°
129 45 36
85 50 42
125 12 22
10 54 49
71 7 7
11 54 55
130 40 27
10. 18 16 5
11. 547 55 0
12. 68 9 43
13. 40 44 44
14. 130 44 44
15. 32° 23 39
9.
15