# Geometry: Measurement of Angles (Level 3 of 9) | Degrees, Minutes, Seconds, Congruent Angles

Measurement of Angles Level 3

In this video we will go over parts of a degree, and congruent angles. Let’s start with parts of a degree. In surveying, navigation and other applications

of angle measure, we often have to measure angles more precisely than to the nearest

degree. We may have to measure angles to the nearest

tenth, hundredth or thousandth of a degree. We can denote fractional parts of an angle

in two ways by using the minutes and seconds notation or by using decimals. In the minutes and seconds notation, a degree

is divided into 60 minutes, for even finer measurements the minute can be divided into

60 seconds. This last unit is very small and it is usually

used for astronomical measurements and measuring latitude and longitude. Now a word of caution, these minutes and seconds

have nothing to do with time they represent smaller and smaller parts of a degree. In the context of geometry degrees minutes

and seconds refer to angle measurements and not units of time. We denote minutes by a single tick mark or

small dash after the number for example 30 minutes is written as follows, seconds are

denoted by using a double tick mark or two small dashes after the number for example

45 seconds is written as follows. The following angle measure would be pronounced

as “67 degrees, 21 minutes, 37 seconds”. When minutes and seconds are used alone, we

usually say “arc minutes” and “arc seconds” to avoid confusion with units of time. A second way of denoting a fractional part

of an angle is by using the degree measure plus a decimal fraction of a degree. Similar to the way we measure time where each

hour of the day is divided into 60 minutes. When we say half an hour we are referring

to 30 time minutes so when we say half a degree we are referring to 30 arc minutes. For example, an angle that measures 2.5 degrees

can be expressed as 2 degrees 30 minutes since 30 minutes represents half a degree. Likewise an angle that measures 2.25 degrees

would be expressed as 2 degrees 15 minutes since 15 minutes represents a quarter of a

degree. In general to convert an angle measure with

a decimal, we simply multiply the decimal by 60 to obtain a number representing the

minutes of a degree, if the minutes happen to also have decimals after multiplying by

60 then we take the decimals of the minutes and multiply them again by 60 to obtain a

number representing the seconds. For example the angle measure 2.135 degrees

can be changed into the degree-minute-second notation by taking the decimal part of the

degree in this case 0.135 and multiplying it by 60 obtaining the number 8.1 this number

represents the number of minutes. Notice that this number has the decimal 0.1

meaning that we have 0.1 of a minute so we take this decimal and multiply it by 60 to

convert it into seconds, doing that we obtain 6 seconds. So 2.135 degrees can be written as 2 degrees,

8 minutes, 6 seconds. At times you might want to convert an angle

measure written in minutes and seconds into a degree measure with a decimal. To convert the angle we simply add the degree

with the fractional representation of the minutes and seconds. For example to convert this same angle written

in minutes and seconds back to degrees with a decimal, we first convert the minutes into

an equivalent degree measure by taking 8 minutes and dividing it by 60 since one degree measures

60 minutes, this way we convert 8 minutes into an equivalent degree measure, we also

need to convert the seconds into an equivalent degree measurement, now, we need to be careful

here, we know that 1 minute is equal to 60 seconds but since we want to convert seconds

into degrees we need to figure out how many seconds are there in 1 degree. Since 1 minute equals 60 seconds and 1 degree

equals 60 minutes then, 1 degree equals 60 times 60 seconds or in this case 3600 seconds. So we convert the seconds into an equivalent

degree measure by taking 6 and dividing it by 3600. Now it is just a matter of adding the degree

measure along with these two fractions, we can use a calculator to approximate the fractions

and add them to 2 degrees doing that we obtain 2.135. Alright, and this is how you deal with parts

of a degree let’s end the video by going over congruent angles. Congruent angles are angles that have the

same measure. In the following diagram the measure of angle

ABC equals the measure of angle DEF, we can write angle ABC is congruent to angle DEF. Notice that when we are referring to the actual

measurement of the angle we use an equal sign and when we are referring to the figures we

use the congruent symbol. The definition of congruent angles tells us

that these two statements are equivalent and we can use them interchangeably. Just like tick marks are used to show that

two line segments are congruent, we use one or more small arc marks to show that two angles

are congruent. For example in the figure shown, triangle

CAT and triangle DOG have various arc marks. In this case angle C and angle D have matching

single arc marks, so these angles are congruent, in the same manner angle A and angle O have

matching double arc marks so these angles are congruent, lastly angle T and angle G

have matching triple arc marks so these angles are congruent. Alright and these are the basics of measuring

angles. We can find the measure of an angle by using

a protractor and the measure of an angle is found by computing the absolute value of the

difference of the degree measurement that the rays or segments correspond with on the

protractor. We can classify angles by sizes and the 4

most common angles are acute, right, obtuse and straight. When we need to measure an angle more precisely

we can use the degrees-minutes-seconds notation to measure fractional parts of a degree. We can also identify and label congruent angles

by using arc marks in a diagram or by using the congruent symbol which tells us that two

angles have the same measurement. Ok, in our next video we will go over simple

examples that makes use of the concepts learned so far.

awesome

Wow that’s a lot of stuff