Download If one side of a triangle is longer than the other side, then the angle

5.5
Use Inequalities in a Triangle
Theorem 5.10:
If one side of a triangle is longer
than the other side, then the
angle opposite the longest side
A
is _______
larger than the angle
opposite the shorter side.
B
8
5
C
AB  BC , so
m ___
C  m ___
A .
5.5
Use Inequalities in a Triangle
Theorem 5.11:
B
If one angle of a triangle is larger
than another angle, then the
side opposite the larger angle
50o
30o
C
A
is _______
longer than the side
mA  mC,
opposite the smaller angle.
so ____
BC  ____
AB .
5.5
Use Inequalities in a Triangle
Example 1 Write measurements in order from least to greatest
Write measurements of the triangle in
order from least to greatest.
a.
C
b.
36o
B
57o
E
22
87o
A
13
F
D
12
Solution
a. m
____
____
A
C mB  m
____
BC
AB  AC  ____
5.5
Use Inequalities in a Triangle
Example 1 Write measurements in order from least to greatest
Write measurements of the triangle in
order from least to greatest.
a.
C
b.
36o
B
57o
E
22
87o
A
13
F
D
12
Solution
b. m
____
____
F
E mD  m
____
DF  EF  ____
DE
5.5
Use Inequalities in a Triangle
Checkpoint. Write the measurements of the
triangle in order from least to greatest.
B
99o
1.
A
34o
47o
C
mA  mC  mB
BC  AB  AC
5.5
Use Inequalities in a Triangle
Checkpoint. Write the measurements of the
triangle in order from least to greatest.
Q
80o
2.
P
45o
55o
R
mP  mR  mQ
QR  PQ  PR
5.5
Use Inequalities in a Triangle
Theorem 5.12: Triangle Inequality Theorem
The sum of the lengths of any
two sides of a triangle is A
greater than the length of
the third side.
___
AB  ___
BC  AC
AC  ___
BC  ___
AB
___
BC
AB  AC  ___
B
C
5.5
Use Inequalities in a Triangle
Example 2 Find possible side lengths
A triangle has one side of length 14 and another of length
10. Describe the possible lengths of the third side.
Solution
Let x represent the length of the third side. Draw diagrams to
help visualize the small and large values of x. Then use the
Triangle Inequality Theorem to write and solve inequalities.
Small values of x
14
x
10
x  10
__  14
__
x  __
4
Large values of x
x
10
14
__  14
__  x
10
__
24
24  x , or x  __
The length of the third side must be
_______________________________.
greater than 4 and less than 24
5.5
Use Inequalities in a Triangle
Checkpoint. Complete the following exercise
3. A triangle has one side 23 meters and another of
17 meters. Describe the possible lengths of the
third side.
Small values of x
Large values of x
23
x
x
17
x  17  23
x6
17
23
17  23  x
40  x , or x  40
The length of the third side must be
greater than 6 meters or less than 40 meters
5.5
Use Inequalities in a Triangle
Theorem 5.13: Exterior Angle Inequality Theorem
The measure of an exterior
angle of a triangle is greater
than the measure of either
of the nonadjacent interior
angles.
m1  m2
____
m1  m3
____
3
2
1
5.5
Use Inequalities in a Triangle
Example 3 Relate exterior and interior angles
Write inequaliti es that relate m1 to mB and mC.
Solution
B and C are nonadjacen t interior angles to 1.
B
So, by the Exterior Angle
Inequality Theorem,
70
_____
m1 60
m1 > 70 and ______>_____.
o
o
1
A
o
60
o
C
5.5
Use Inequalities in a Triangle
Checkpoint. Complete the following exercise
4. Write inequaliti es that relate m1 to
mA and mB.
B
1
o
38
C
112
A
o
A and B are nonadjacen t interior angles to 1.
m1  112
o
m1  38
o
5.5
Use Inequalities in a Triangle
Pg. 299, 5.5 #1-23