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Geometry
Lesson 4 – 2
Angles of Triangles
Objective:
Apply the triangle Angle-Sum Theorem.
Apply the Exterior Angle Theorem.
Triangle Angle-Sum Theorem
Triangle Angle-Sum Theorem

The sum of the measures of the angles of a
triangle is 180.
Auxiliary line
Auxiliary line – an extra line or segment
drawn in a figure to help analyze
geometric relationships.
Given : ABC
Pr ove :
m1  m2  m3  180
Find the measure of each
numbered angle.
Which angle do we find first?
Could find angle 2 first since we
Know 2 other angles in that triangle.
Or we could find angle 1 since we
Know it forms a linear pair with 57.
m2  57  71  180 m3  m1  28  180
m2  128  180 m3  123  28  180
m2  52
m3 151  180
m1  57  180
m3  29
Since linear pair. Don’t
m1 123 get this 180 confused
with adding all the angles
of a triangle to 180.
Find the measure of each
numbered angle.
m7  58  65  180
m7 123  180
m7  57 m5  57
m6  m7  180
m6  57  180
m6  123
m4  m5  67  180
m4  57  67  180
m4  124  180
m4  56
m6  m8  m9  180
123  m8  m8  180
2(m8)  57
m8  28.5
m9  28.5
Exterior & Remote interior angles
Exterior angles of a triangle:
Formed by one side of the triangle and the
extension of an adjacent side.
 Exterior angle that is adjacent to the triangle.

Remote Interior angles

The 2 interior angles inside the triangle that are not
adjacent to the exterior angle.
Name an exterior angle and the pair
5 of remote interior angles that go with it.
4
6
3
12
11
1
10
Exterior
7
2
9
8
4
6
7
9
10
12
Remote Int.
1 & 2
1 & 2
1& 3
1& 3
2 & 3
2 & 3
Angles 5, 8, and 11 are not exterior angles since they are
not adjacent to a side of the triangle.
Theorem
Exterior Angle Theorem

The measure of an exterior angle of a
triangle is equal to the sum of the
measures of the two remote interior
angles.
Find the measure of angle FLW.
2x – 48 = x + 32
x – 48 = 32
x = 80
Exterior angle = Sum of Remote Interior angles
FLW  2 x  48  2(80)  48  112
Corollary
Corollary –

A theorem with a proof that follows as a
direct result of another theorem.
Corollaries
Corollary 4.1

The acute angles of a right triangle are
complementary.
Corollary 4.2

There can be at most one right or obtuse
angle in a triangle.
Find the measures of each
numbered angle.
m1  90  52  180
m1 142  180
m1 38
m2  90  38  180 or m2  38  90
m2  128  180 m4  m3  90  180
m2  52
m4  38  90  180
m4  128  180
m3  m2  90
m4  52
m3  52  90
m3  38
Find the measures of each
numbered angle.
m5  41  90
m5  49
m3  48  90
m3  42
m2  105  180
m2  76
m1 104
104
56
m4  m2  56  180
m4  76  56  180
m4  132  180
m4  48
Homework
Pg. 248 1 – 11 all, 12 – 40 EOE,
50, 60, 64, 66