Measurement of Angles Level 1

In the following series of videos we will learn how to measure angles, classify angles

by size, name the parts of a degree and recognize congruent angles. In this video we will review how to name angles,

we will then go over how to measure angles. In the previous videos we define an angle

as a figure formed by two line segments or rays meeting at a common point called the

vertex the plural would be vertices. The segments or rays that form the angle are

called the sides of the angle. Angles are named by using the angle symbol. You can name an angle in a couple of ways

we can use a vertex, we can use a point located on each ray or line segment and the vertex,

or we can also use a single number. Another common way to name angles is by using

Greek letters such as theta, alpha, or beta instead of numbers. For example the following angle can be denoted

as angle SRT, or angle TRS or angle R or angle 1 or theta. The set of all points between the sides of

the angle is the interior of the angle. Here the word interior is Latin for “inner”. The interior of an angle is the area between

the two rays or line segments that define it, the sides of the angle form “jaws”

that extend out to infinity. The exterior of an angle is the set of all

points outside the angle, in other words the region on the plane that is not the interior. Now that we have reviewed the basics of angles,

let’s talk about how we measure them. Just like a ruler is used to measure a line

segment a protractor is an instrument that is commonly used to measure angles. Angles are usually measured in terms of degrees,

radians, grads, or nautical angles. In this course we will be using degrees as

the unit for measuring angles. You can think of the measure, or size, of

an angle as the amount of turning you would do if you were at the vertex, looking along

one side of the angle, and then turned to look along the other side of the angle. If you turned all the way around to face your

starting direction you would turn 360 degrees, meaning that you turned around in a complete

circle. We will cover the properties of circles in

greater detail in a much later video, for now we will use a circle to visually represent

the idea of turning around, this is why we use a circle as a symbol to denote angles. We denote angles by using the degree symbol

which is represented by a small raised circle that floats above the right side of the number

just like an exponent. Now that we know that if we turn around in

a complete circle we would turn 360 degrees, the next thing to understand is the size of

a single degree. To get a sense of the size of a single degree,

let’s take a circle and slice it into 360 equal pieces by using straight cuts that go

through the center of the circle, by doing this we end up with 360 individual angles

and each angle or slice would measure 1 degree. So a degree is equivalent to 1/360 of a complete

revolution around the circle. So a protractor is nothing more than half

a circle broken up into 180 equal slices. A typical protractor will have two set of

numbers, the inner numbers start at 0 degrees which is located at the lower right edge of

the protractor and increase as you move counterclockwise around the protractor, the outer numbers start

at 0 degrees which is located at the lower left edge of the protractor and increase as

you move clockwise around the protractor. We measure angles by placing the center mark

of the protractor on the vertex of the angle and we align one ray (or segment) of the angle

with the 0 degree mark at either side, then the measure of the angle is given by the number

that falls on the other ray (or segment). We usually use the inner numbers on the protractor

for angles measured counterclockwise, and we use the outer numbers for angles measured

clockwise. For example the measure of angle R is 30 degrees,

similar to the measurement of a line segment; the measurement of an angle is denoted in

a distinct manner. In order to denote the measurement of an angle

we first write a lower case m followed by the name of the angle, this is read as “the

measurement of angle R is 30 degrees”. At times when the context is clear we can

go ahead and denote the measurement without writing the lower case m, this is rarely done

in most geometry textbooks. In these videos we will be using the lower

case m to denote the measurement of an angle. If we have multiple angles that share the

same vertex like the following figure we need to use 3 letters to denote each of the angles. In this example angle ABC measures 60 degrees,

this angle measure is equivalent to 1/6 of a revolution around a circle, angle ABD measures

90 degrees and this is equivalent to 1/4 of a revolution around a circle, angle ABE measures

120 degrees and is equivalent to 1/3 of a revolution around a circle, angle ABF measures

180 degrees and it is equivalent to 1/2 a revolution around a circle. Some math courses deal with negative angles,

zero angles, and angles greater than 180 degrees. In this course we will be working with angles

that are greater than 0 degrees and less than or equal to 180 degrees. When using a protractor it is not necessary

to always align one of the sides of the angle with the 0 degree mark to measure the angle

between 2 rays or 2 segments. Similar to the way we measured a line segment

by using the coordinates of the endpoints, the measure of an angle is the absolute value

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

on the protractor. For example the measurement of angle EBC can

be found by taking the absolute value of the difference between 120 degrees and 60 degrees

or the absolute value of the difference between 60 degrees and 120 degrees. In this case the measurement of angle EBC

is 60 degrees. In general the measure of an angle is the

absolute difference of the real numbers that the rays or segments correspond with on the

protractor. For example if ray BA corresponds with the

real number A and ray BC corresponds with the real number C then the measurement of

angle ABC would be equal to the absolute difference of A minus C or the absolute difference of

C minus A. Alright in our next video we will learn how

to classify angles by sizes.

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