Notes on Triangles and Angles (1)
Name_________________________
A vertex of a polygon or an angle is the point at which two sides of a polygon, or the two sides
of an angle, intersect. The plural of the word "vertex" is "vertices".
The interior angles of a triangle are the three angles formed inside of a triangle at each of the
three vertices.
The Interior Sum Theorem for triangles states that the sum of the measures of the three interior
angles of a triangle always equals 180 .
-------------------------------------------------------------------------------------------------------------------1. In ABC , the measure of A  70 , and
2. Find the measure of R in the
mB  50 . What is mC ?
figure below.
T
131
R
25
S
--------------------------------------------------------------------------------------------------------------------In a picture or situation, when a point serves as the vertex of only one angle, then it is
appropriate to use only the letter of the vertex to name the angle. For example, in #2 above, the
interior angle which measures 131 is the only angle with a vertex at T. Thus, that angle can be
named T .
N
However, in the figure to the left, there are two angles formed at
vertex P (angles 1 and 2). Since this is the case, if a person
wishes to refer to angle 1, then he/she must use three letters to
name the angle.
1 P
2
M
Angle 1 could also be called NPM . What would be the other
appropriate three-letter name for Angle 1?
--------------------------------------------------------------------------------------------------------------------In the figure above, note that side NP has been extended in order to form angle 2 on the outside
of the triangle at vertex P. Angle 2 is known as an exterior angle.
An exterior angle of a triangle is an angle that forms a linear pair with one of the interior angles
of a triangle.
It is important to remind you of the definition of a linear pair - two angles that share the same
vertex and one side, but whose different sides form a line, line segment, or ray.
In the space to the right, draw a horizontal line. In the "middle" of the line,
place point D. Now, construct a ray from point D that goes upward and to
the right. Two angles have been formed - label them as 3 and 4. They form
a linear pair.
As you may recall from middle school, if a line could be given an angle measure, the measure of
the line would be ___________ degrees.
Consider the case of the linear pair. It is two angles that make up a line. Thus, the sum of the
measures of the two angles in a linear pair must equal ___________ degrees.
N
Finally, recall this picture from the front. Cover up MN if
necessary to clearly see the linear pair formed.
1
Since 1 and 2 are the two angles which make up a linear
pair, then their measures must add to equal 180 .
P
2
M
Hence, the sum of the measures of an exterior angle of a
triangle and the interior angle from the same vertex is 180 .
--------------------------------------------------------------------------------------------------------------------In the figure to the left, construct an exterior angle to DEF at
vertex F. Label the angle with the number 5.
E
D
Did everyone construct the same exterior angle? What can be
learned from this?
F
Now, what would be the appropriate name for the interior
angle at vertex F? Label the angle with the number 6.
Complete the statement: m5  _________  180 .
--------------------------------------------------------------------------------------------------------------------Let's summarize the learning to this point using the figure
to the right.
m1  m2  m3  _________ .
2
1
3
4
Also, m3  m4  _________ .
Freeze! If both of these quantities above equal 180 , then they must be equal to each other!!!
Thus, m1  m2  m3  m3  m4 ! However, since the quantities on both sides of the
equation have m3 , it can be subtracted from both sides, leaving:
m1  m2  m4
Look at the figure above to consider the implications of this statement. In a triangle, the two
interior angles that are nonadjacent to the exterior angle are called remote interior angles.
Continue to next page.
Notes on Triangles and Angles (2)
Name_________________________
In the figure, 1 and 2 are the remote interior angles in
relation to 4 . In addition, as proven on page one of the
notes, m1  m2  m4 .
If all of this is put together, then the resulting theorem can
be formed:
2
1
3
4
Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of
the measures of the remote interior angles.
Based on this theorem, there is another obvious statement:
Exterior Angle Inequality Theorem: The measure of the exterior angle of a triangle is greater
than either of the remote interior angles.
--------------------------------------------------------------------------------------------------------------------N
Let's once again use the figure to the left.
1
Suppose mM  72 , and angle 2 measures 127 . What is the
measure of angle N?
P
2
M
Well, consider that M and N are the remote interior angles to
exterior angle 2. Thus, let's use the Exterior Angle Theorem from
above.
mM  mN  m2
72  mN  127
mN  55
--------------------------------------------------------------------------------------------------------------------Homework on Triangles and Angles
For Questions 1-7, name the angle numbered the same as the question number using the
appropriate one-letter or three-letter label. The point in the center of all of the segments is
labeled point H. For Question 8, name the large angle formed by angles 1 and 2 combined.
D
3
C
A
1
6
2
H
4
5
F
G
B
7
E
I
1.
2.
3.
4.
5.
6.
7.
8.
9.
Triangle UVW is a triangle in which mV  10 and mW  20 . What is mU ?
10.
A right triangle has one interior angle that measures 40.5 . Provide the measures of the
other two interior angles of the triangle.
--------------------------------------------------------------------------------------------------------------------An exterior angle has been drawn in the figure to the left. Use the
M
figure for Questions 11-13.
J
x
2x
11.
What is the value of x?
12.
What are the names of the two remote interior angles of
the exterior angle drawn?
13.
Complete the statement: mK  mL  m _________
K
L
--------------------------------------------------------------------------------------------------------------------14.
Do the two exterior angles which can be formed from the same vertex of a triangle have
the same measure? Think carefully about the Exterior Angle Theorem.
--------------------------------------------------------------------------------------------------------------------Consider for Questions 15-16: An exterior angle of a triangle has a measure of 103 .
15.
What is the measure of the interior angle with the same vertex?
16.
If one remote interior angle measures 59 , what is the measure of the other remote
interior angle?
--------------------------------------------------------------------------------------------------------------------17.
A triangle has interior angles which measure 89 , 54 , and 47 . What is wrong with
this statement?
--------------------------------------------------------------------------------------------------------------------18.
Consider the figure to the left. Which of the angles listed
I
below has the greatest measure?
G
A) F
D) IGH
70
B) FGH
E)There is not enough information.
C) H
F
H
--------------------------------------------------------------------------------------------------------------------A triangle has an exterior angle. The two remote interior angles measure 64 and 41 .
19.
What is the measure of the exterior angle?
20.
What is the measure of the other interior angle?