Download Triangle Inequality Theorem

Triangle
Inequalities
1
Triangle Inequality

The smallest side is across from the smallest angle.
A is thesmallest angle,  BC is thesmallest side.

The largest angle is across from the largest side.
B is the largest angle,  AC is the largest side.
B
89
54
C
37
A
2
Triangle Inequality – examples…
For the triangle, list the angles in order from least to greatest measure.
B
A
AB is the smallest side  C smallest angle.
C
5 cm
BC is thelargest side  Ais the largest angle.
Angles in order from least to greatest  C , B, A
3
Checkpoint: Use Inequalities in a Triangle
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 .
Use Inequalities in a Triangle
B
If one angle of a triangle is larger
than another angle, then the
side opposite the larger angle
50o
A
is _______
longer than the side
mA
opposite the smaller angle.
30o
C
 m  C,
so ____
BC  ____
AB .
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  ____
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
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
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
Triangle Inequality – examples…
For the triangle, list the sides in order from shortest to longest measure.
(7x + 8) ° + (7x + 6 ) ° + (8x – 10 ) ° = 180°
B
(8x-10)◦
22 x + 4 = 180 °
22x = 176
m<C = 7x + 8 = 64 °
X=8
m<A = 7x + 6 = 62 °
54 °
m<B = 8x – 10 = 54 °
A
B is the smallest angle  AC shortest side.
62 °
(7x+6)◦
64 °
C
(7x+8)◦
C is the largest angle  AB is thelongest side.
Sides in order from smallest to longest  AC , BC , AB
10
Use Inequalities in a Triangle
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
Triangle Inequality Theorem:
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side. B
a+b>c
a
c
a+c>b
b+c>a
A
C
b
Example: Determine if it is possible to draw a triangle with side
measures 12, 11, and 17. 12 + 11 > 17  Yes
Therefore a triangle can be drawn.
11 + 17 > 12  Yes
12 + 17 > 11  Yes
12
Finding the range of the third side:
Since the third side cannot be larger than the other two added
together, we find the maximum value by adding the two sides.
Since the third side and the smallest side cannot be larger than the
other side, we find the minimum value by subtracting the two sides.
Example: Given a triangle with sides of length 3 and 8, find the range of
possible values for the third side.
The maximum value (if x is the largest The minimum value (if x is not that largest
side of the triangle)
3+8>x
side of the ∆)
8–3>x
11 > x
5> x
Range of the third side is 5 < x < 11.
13
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
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
Use Inequalities in a Triangle
Exterior Angle Inequality Theorem
The measure of an exterior
angle of a triangle is equal
to the sum of the two non
adjacent interior angles.
3
2
1
m1  m2  m3
Use Inequalities in a Triangle
Example 3 Relate exterior and interior angles
Solution
 B and  C are nonadjacen t interior angles to 1.
B
So, by the Exterior Angle
Inequality Theorem,
m1  60  70
m1  130
70
1
A
o
60
o
C
Use Inequalities in a Triangle
Checkpoint. Complete the following exercise
Find the measure of angle 1.
B
38
1
o
C
112
A
o
m1 150
o