The Mathematics 11
Competency Test
Angles and Angle Measurement
When two lines or line segments intersect in a plane, they form angles.
The figure to the right shows such a situation. The lines forming the
angle are called its sides, and the point at which the lines meet is called
the vertex of the angle.
A
B
Another way to view the formation of an angle is by starting with a ray
and rotating it about its endpoint. The angle is then a measure of the
amount of rotation that occurs to get from the initial position of the ray to
its final position. (Hence the two sides of any angle are called its initial
side and its terminal side.)

C
Angles are normally drawn as arcs, sometimes with arrowheads to indicate direction. Sometimes
the arc itself is labelled with a literal symbol (the Greek letter  or “theta” is a favourite for this). It
is also common to refer to the angle in the figure using the notation ABC or CBA. Sometimes
people will refer simply to “angle B.” Without a labelled sketch, these notations can be
ambiguous since even in the simple situation shown in the figure above, there are two possible
angles to which the symbols “B” or “ABC” could be referring: the smaller angle shown, and the
larger angle formed by thinking of the rotation being clockwise from BC to BA.
Angle Measurement
The two most common units of measurement for angles are degrees and radians. There are
two variations used in the “degree system” – here we describe only the one called degree
decimal.
In the degree decimal system
1 complete revolution = 360 degrees or 3600.
Degrees can be symbolized by a superscript zero after the number. Parts of a degree are
indicated by digits to the right of the decimal point as is done for other quantities in life. Thus, the
angle 72.530 has 720 plus 0.53 of a 73rd degree. Since 72/360 = 1/5, the angle 72.530 is just
slightly more than one-fifth of a complete revolution.
For the BCIT Mathematics 11 Competency Test, all problems involving angles use the degree
decimal system for expressing the size of angles.
Other Systems of Angle Measurement
As mentioned, there are two other systems of angle measurement used in various applications.
In the radian system,
1 complete revolution = 2 radians  6.28 radians
Here  is the familiar ratio between the distance around a circle and its diameter. This may seem
like a rather silly way to come up with a unit of measurement for angles, but there are some
significant advantages to using radians as units of measurement for angles when you are dealing
with problems involving higher mathematics.
David W. Sabo (2003)
Angles and Angle Measurement
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In the degree-minute-second or DMS system,
1 complete revolution = 360 degrees,
but parts of a degree are expressed in terms of so-called minutes and seconds:
1 degree = 60 minutes
1 minute = 60 seconds
This system is still used by land surveyors, cartographers, astronomers, etc.
The reason you need to know about these alternative unit systems for measuring angles is that
your calculator will have a key for switching between these systems and perhaps others as well.
So, if the calculator is set in radian mode (usually indicated by the abbreviation “rad” somewhere
on the display), then it will assume all angles are being expressed in units of radians. If you are
intending that the angles you key in to the calculator are in units of degrees, you will get the
wrong answer in problems involving angles. So, before calculating angle values, or before
entering angle values into calculations of other things, (as will happen in the section of these
notes covering trigonometry), make sure that your calculator is set to the appropriate angle
measurement mode for your problem.
Special Angles:
Some angles have special names:
A right angle corresponds to ¼ of a complete revolution. Right angles are
denoted by small boxes rather than circular arcs. Lines that meet at right
angles are said to be perpendicular. Obviously, the measure of a right
angle is 900.
A straight angle looks like no angle at all, but corresponds to
one-half rotation or 1800.
Acute angles are angles which measure between 00 and 900. They are
narrower than a right angle.
Obtuse angles are angles which measure between 900 and 1800.
They are broader than a right angle, but not as broad as a straight
angle.
David W. Sabo (2003)
Angles and Angle Measurement
right
angle
straight angle
acute
angle
obtuse
angle
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Two angles are said to be complementary if they add up to 900 or a right
angle. In the figure to the right, angles A and B are complementary.
B
A
Two angles are said to be supplementary if they add up to 1800
or a straight angle. In the figure to the right, angles A and B are
supplementary.
David W. Sabo (2003)
Angles and Angle Measurement
B
A
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