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.