1. Unit 20
ANGULAR MEASURE

2. 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°

3. 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

4. EXPRESSING DEGREES, MINUTES, AND SECONDS AS DECIMAL DEGREES
Converting to decimal degrees
We can use unity fraction method or dimensional analysis like in units 8 & 9
Start with the seconds and convert to minutes.
Then convert the minutes into degrees and add the decimal onto your degrees.
EXPRESSING DEGREES, MINUTES, AND SECONDS AS DECIMAL DEGREES

5. CONVERSION EXAMPLE Express 73°5748 as decimal degrees:
Convert the 48 to minutes
Add that to your minutes
So 57.8’
Now change that to degrees
Add to the 73°
So °

6. CONVERSION EXAMPLE Convert the decimal to minutes So we have 32’
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
So we have 32’
Now convert that decimal minutes to seconds
So 48° 32’ 24”

7. ADDING ANGLES Add the following angles: Simplify the sum: 49° 53 37
49° 53 37
+ 38 47 24 Add seconds to seconds, minutes to minutes, etc.
Simplify the sum:
87° 100 61
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

8. 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.

9. Subtract the following 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
SUBTRACTING ANGLES
88 82 75
– 70 35 20
18 47 55 Ans

10. MULTIPLYING ANGLES Multiply 51 33 42 by 3: 51 33 42  3
 3
Simplify the product
153° 99 126
126 = 2 6
Adding these 2 to the 99, we now have 101  = 1 41
Adding this degree to the 153 degrees, we now have 154°
Combining units, we now have 154 41 6 Ans

11. DIVIDING ANGLES Divide 147 55 34 by 2: 73 57 47 Ans 146 1°= 60
+ 55
115
114
1= 60
+ 34
94
94

12. COMPLEMENTS AND SUPPLEMENTS
Two angles are complementary when their sum is 90°
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
COMPLEMENTS AND SUPPLEMENTS
89 60
– 43 18
46 42 Ans

13. PRACTICE PROBLEMS 143 54 32 242 33 24 129.76° 85.845°
Express the following degrees, minutes, and seconds as decimal degrees. Round to three decimal places where necessary:
143 54 32
242 33 24
Express the following decimal degrees as degrees, minutes, and seconds:
129.76°
85.845°
Perform the following operations. Simplify all answers:
45° 54 39  17 43
87 16 25 – 76 21 36

14. PRACTICE PROBLEMS (Cont)
54 47 32  19 35
72 15 15 – 60 20 20
43 33 29  3
54 48 15  3
136 58 45  4
272 38 52  4
Determine the complement of 49 15 16
Determine the supplement of 49 15 16
Determine the supplement of 147 36 21

15. PROBLEM ANSWER KEY 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
18 16 5
547 55 0
68 9 43
40 44 44
130 44 44
32° 23 39