Angles
Intro
An angle is the union of two rays which meet at a common point, called the vertex.
Angles are formed when lines, rays or line segments intersect. Angles can be classified by their
measure.
Type of Angle
Degree Measure
Acute
Less than 90°
Right
Exactly 90°
Obtuse
More than 90°
Straight
Exactly 180°
Reflex
Example
Between 180° and 360°
Naming Angles
There are several ways to name an angle. For example, you can use three points to name an
angle; just remember the vertex must always be the middle point.
This angle can be written as ABC or
CBA. Notice the vertex in either case is
always the middle point.
A
B
C
 is the symbol for angle.
This angle can also be named as B, however, only use this notation when there are no other
angles that share B as the vertex.
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Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
1
Angles which share a common vertex are called adjacent angles. Take a look at the following
examples.
A
1
B
2
Z
C
In the example on the left, AZB and BZC are adjacent angles because they share a common
vertex, Z. However, the example on the right shows two angles, 1 and 2, which are not
adjacent because they do not share a common vertex.
Equivalent Angles
You may encounter angles which look like this:
mA = mB
This means angles A and B are equal in measure (the little “m” means “measure of”).
The notation is different than A  B which means angles A and B are congruent.
When the measure of two angles is the same, you use the “equal (=)” sign because you are
comparing two real numbers.
If mA°= 35 and mB = 35°, then mA = mB
If you are talking about two angles being exactly the same, you use the “congruent ()” symbol
because you are comparing two figures.
A  B
For example,
A
mAZB = 60°; mBZC = 60°; mAZC = 120°
B
AZB  BZC
60°
60°
Z
4
C
Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
2
Special Pairs of Angles
Two angles whose sum is 90° are called complementary angles.
The sum of these two adjacent angles is 90°, therefore
they are complementary angles.
50°
40°
The sum of these two angles is 90°, therefore they are
complementary angles. We also say these angles are
complements of each other.
60°
30°
135°
G
D
H
45°
45°
The sum of GHF and DHE is 90°, therefore they are
complementary angles. We can also say GHF is the
complement of DHE.
E
135°
F
Two angles whose sum is 180° are called supplementary angles.
The sum of these two adjacent angles is 180°, therefore
they are supplementary angles. We also say that each
angle is the supplement of the other.
135°
45°
The sum of these two angles is 180°, therefore they are
supplementary angles. We also say that each angle is the
supplement of the other.
30°
150°
4
Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
3
Linear Pairs
Two adjacent angles whose non-shared side is a straight angle are called a linear pair.
A linear pair is always supplementary; meaning the sum of their measures is 180°.
1
1
2
Linear Pair
2
Not a linear pair.
For example,
The figure shows a linear pair. To find the
measure of angle 2, subtract 140° from 180°
(180° - 140° = 40°).
140°
2
m2 = 40°
Vertical Angles
When two lines or line segments intersect, they form vertical angles.
Vertical angles (also called opposite angles) are congruent, that is, they have the same angle
measure.
2
3
1
4
In this figure, 2 and 4 are vertical angles. 1 and 3 are
also vertical angles.
2  4 and 1  3
For example, in the figure above, if the m4 = 55°, then the m2 = 55° since they are vertical
(or opposite) angles.
Moreover, if the m1 = 35°, then the m4 = 145° because 1 and 4 are a linear pair.
4
Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
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Angles Formed by Transversals
A transversal is a line which intersects two (or more) lines at two (or more) points.
1 2
3 4
m
6
5
7
l
8
t
In this figure, the transversal is line t. Notice when a transversal intersects two lines, it forms
eight angles. Each of these angles has a special name; you can look up the names of these
angles in book or online.
When a transversal intersects a pair of parallel lines, these special angles also have special
relationships.
t
1
3 4
5 6
7 8
2
l
m
Each pair of angles in the diagram has a theorem or postulate associated with it. Instead of
memorizing each one of them, just remember the following:
If the pair of angles looks the same (e.g., both acute or both obtuse), they are congruent. If
they look different, they are supplementary.
Now, I know what you are thinking! Not very mathematically sound, is it? However, after years
and years of teaching this stuff, I have found the “layman’s” statement above works wonders.
If you would like a more concise definition of why this works, check out:
http://www.mathopenref.com/transversal.html.
Remember, the layman’s statement only works when a transversal intersects parallel lines.
For example, 1 and 8 are obtuse, therefore they are congruent.  3 and 5 are different
(one is acute, one is obtuse) therefore they are supplementary.
4
Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
5
Angle Problems
Example 1
B
Given: DCE is a complement of BCA.
Find the m1 and m2.
D
75
°
Note: Figure not drawn to scale.
1
2
Solution
65
C°
A
E
Since DCE and BCA are complements, the sum of their measure is 90°.
Therefore, m2 = 25° (90° - 65°).
To find m1, we know ACE is a straight angle which means its measure is 180°. Since we
know m2 = 25° and mDCE = 65°, then m1 = 90° (180° - 25° - 65°).
Notice the diagram does not make sense since the m1 = 90° and our diagram is not drawn correctly. However,
since we have a note that states the figure is not drawn to scale, we are ok. This is important because most
assessments will have diagrams which are not drawn to scale. So, when this happens, do not try to measure
angles with your protractor because you will get an incorrect solution.
Example 2
143°
Find the measures of the missing angles.
a
b
Solution
Since mb and the given angle are vertical angles, they are congruent which means mb = 43°.
The given angle and a are a linear pair, therefore they are supplementary.
As such, ma = 37° (180° - 143°).
Example 3
Given: l || m
Find the value of x.
54°
9x
l
m
Solution
Since line l is parallel to line m, our layman’s statement will hold true. Therefore, we can say
9x = 54°, so x = 6.
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Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
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