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.