Triangle Inequalities
Part 2
Exterior Angle Inequality Theorem
• If an angle is an exterior angle of a triangle
then its measure is greater than the measure
of either of its corresponding remote interior
angles.
2
m4 > m1
1
3
4
m4 > m2
Example
Which angles are less than
m3?
D
m3 > m 4
6
m3 > m 6
C
5
3
Why is m 2 > m 6?
4
2
Since m 2 = m 3 becuase
of the converse of the
isosceles trainge theorem,
then m 2 must also be
greater than m 6
A
1
B
Triangle Sides
• If one side of a triangle is longer than another side,
then the angle opposite the longer side has a greater
measure than the angle opposite the shorter side.
B
C is the largest angle
B is the middle angle
5
9
A is the smallest angle
C
A
7
Examples
• List the Angles from largest to smallest
Largest P
Largest C
Medium M
Medium B
Smallest N
Smallest A
Triangle Angles
• If one angle of a triangle has a greater
measure than another angle, then the side
opposite the greater angle is longer than the
side opposite the lesser angle.
E
DFis the largest side
77°
DEis the middle side
65°
D
38°
F
EFis the smallest side
Examples
• List the sides of triangle from largest to
smallest
Largest QS
Largest CB
Medium QR
Medium CA
Smallest RS
Smallest AB
Triangle Inequality Theorem
• The sum of the lengths of any two sides of a
triangle is greater than the length of the third
side.
– Always check using the two smallest sides, they
must be larger than the third. If this is true the
numbers will represent a triangle.
Example
• Do these numbers represent a triangle?
1.) 9, 7, 12
Yes
2.) 5, 5, 10
No
3.) 1, 4, 6
No
4.) 6, 6, 2
Yes
Finding Range of Third Side
• If you are given two sides of a traingle you can
determine the range that the third side must
fall in.
– To find the smallest possible side length you
subtract the larger side from the smaller side. The
value you get can not be a side, however
everything larger will work
– To find the largest your third side could be you
add your two given sides, although this value will
not work everything less than it will.
Example
• If you have two sides of a triangle 4 in and 7 in
what is the range for the possible third side, n.
3in < n < 11in
• If you have two sides of a triangle 8 in and 12
in what is the range for the possible third side,
n.
4in < n < 20in
Hinge Theorem
• If two sides of one triangle are congruent two
two sides of another triangle, and the
included angles are not congruent, then the
longest side is opposite the larger included
angle.
If m B > m X
Then AC > DF
D
A
F
E
C
B
Example
What is the relationship between AC and DE?
C
7
B
E
123 °
4
4
A
78 °
D
7
AC > DE
F
Converse of the Hinge Theorem
• If two sides of one triangle are congruent to
two sides of another triangle, and the third
sides are not congruent, then the larger
included angle is opposite the longer third
side.
If AC > DF
Then m
B > m X
D
A
F
E
C
B
Example
If AC > DE , what is the relationship between
B and F?
C
7
B
E
4
4
A
F
D
mB > m F
7