Angle Relationships
Complementary, Supplementary, and Congruent Angles
Supplementary angles
are angles that add up to
180°. The angles below are
supplementary.
Complementary angles
are angles that add up
to 90°. The angles below
are complementary.
36° + 54° = 90°
Congruent angles are angles
that are equal to each other.
The angles below are
congruent.
79° + 101° = 180°
Adjacent Angles
Linear Pairs
“Adjacent” means “next to”. Adjacent angles are
next to each other. More specifically, they share a
common side between them and a common vertex.
Linear pairs are a special case of adjacent
angles. They are adjacent angles that form a straight
line together.
y
These angles do not form a
linear pair because the line they
form together is not straight.
Because linear pairs form a straight line, they
add up to 180° which makes them supplementary.
Note that all linear pairs are adjacent and
supplementary but not all adjacent and supplementary
angles are linear pairs!
Vertical Angles
Angles that are on opposite sides of two
intersecting lines are called vertical angles. The
angles must be formed by the same lines in order to
be vertical.
Vertical angles are always congruent (equal).
To see why, consider the figure below. Angles c
and a form a linear pair, so they must add up to
180°. Since angle c is 50°, angle a must be 130°.
But angles a and b also form a linear pair, so if
angle a is 130°, angle b must be 50°, just like
angle c is!
p
Angles c and a form a linear pair.
Angles a and b form a linear pair.
Angles c and b are vertical.
Writing Equations to Describe Diagrams
Writing Equations
\ You might encounter problems for which you will have to write equations that describe diagrams.
Some key things to remember when doing this is that angles that form a right angle together add up to 90°,
angles that form a straight line together add up to 180°, and angles that form a full rotation together add up to
360°.
a
71°
a
71°
In the diagram above, angle a
and the 71 degree angle form a
right angle together, so we know
they must add up to 90°. Here
are some equations we could
write…
In the diagram above, angle a
and the 71 degree angle form a
straight line together, so we
know they must add up to 180°.
Here are some equations we
could write…
𝑎 + 71 = 90
71 + 𝑎 = 90
𝑎 + 71 = 180
71 + 𝑎 = 180
90 − 71 = 𝑎
90 − 𝑎 = 71
180 − 71 = 𝑎
180 − 𝑎 = 71
71°
a
In the diagram above, angle a and
the 71 degree angle form a full
rotation (circle) together, so we
know they must add up to 360°.
Here are some equations we could
write…
𝑎 + 71 = 360
71 + 𝑎 = 360
360 − 71 = 𝑎
360 − 𝑎 = 71
When an Unknown Quantity Does Not Represent an Angle
Sometimes an unknown quantity written in a diagram does not represent an angle. If it is part of an
expression, it must be solved for and the value plugged back into the expression to determine the angle
measure.
Example: What are the measures of the angles in the following diagram?
6𝑗 − 2
2𝑗 + 32
30°
Solution:
Begin by writing an equation that describes the
diagram, then solve it.
2𝑗 + 32 + 6𝑗 − 2 + 30 = 180
8𝑗 + 60 = 180
−60 − 60
8𝑗
= 120
÷8
÷8
𝑗
= 15
Next, plug the value you got back into the
original expressions.
2𝑗 + 32
6𝑗 − 2
2(15) + 32
30 + 32
62
6(15) − 2
90 − 2
88
The angles represented by the expressions with
j are 62° and 88°.