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B
A
C
B
2
1
A
3
C
B
2
1
A
3
C
Angles 1, 2, and 3 are interior angles of the
triangle.
B
2
1
A
3
4
C
Angle 4 is an exterior angle of the triangle.
7
8
A
9
1
B
2
6
4
3
5
C
Angles 4, 5, 6, 7, 8, and 9 are exterior angles of
the triangle.
7
8
A
9
1
B
2
6
4
3
5
C
Angles 4, 5, 6, 7, 8, and 9 are exterior angles of
the triangle.
Every triangle has 3 interior angles and 6
exterior angles.
7
8 1
A
9
B
2
6
3 4
5 C
All 6 exterior angles of an acute triangle are
obtuse.
B
7 2 6
3
A
8
9
1
4
5
C
An obtuse triangle has 4 exterior angles that are
obtuse and 2 exterior angles that are acute.
B
8
9 2
7 1
A 6
3
4
5
C
An right triangle has 4 exterior angles that are
obtuse and 2 exterior angles that are right.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
7
8 1
A
9
B
2
6
3 4
5 C
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
7
8 1
A
9
B
2
6
3 4
5 C
Angles 1 and 2 are the remote interior
angles with respect to angles 4 and 5.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
7
8 1
A
9
B
2
6
3 4
5 C
Angles 1 and 3 are the remote interior
angles with respect to angles 6 and 7.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
7
8 1
A
9
B
2
6
3 4
5 C
Angles 2 and 3 are the remote interior
angles with respect to angles 8 and 9.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
B
A
C
•D
We will prove that the measure of
is greater than the measure of
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
B
M
A
•
C
•D
We start by locating and labeling (M) the
midpoint of BC.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
B
M
A
•
C
Draw ray AM.
•D
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
P
B
•
M
A
•
C
•D
Locate and label point P on AM so that
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
P
B
•
M
•
C
A
Draw CP.
•D
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
P
B
•
M
•
A
ABM ~
=
C
•D
PCM by SAS.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
P
B
•
M
•
A
ABM ~
=
C
•D
PCM by SAS.
by CPCTC.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
P
B
•
M
•
A
C
•D
Angle addition tells us that the
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
P
B
•
M
•
A
C
•D
Angle addition tells us that the
Therefore
by substitution.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
P
B
•
M
•
A
C
•D
Drawing BP gives us two new congruent triangles
that will help prove that the exterior angles at
vertex B have greater measures than
.
Theorem 30: The measure of an exterior
angle of a triangle is greater than the
measure of either remote interior angle.
P
B
•
M
•
A
C
•D
Repeating the proof by locating the midpoints
of the other two sides will prove it for the rest
of the angles.
In this lesson we introduced
interior and exterior angles of
triangles.
In this lesson we introduced
interior and exterior angles of
triangles.
Theorem 30: (The exterior angle
inequality theorem)
The measure of an exterior angle of
a triangle is greater than the measure
of either remote interior angle.