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5.5 Inequalities in Triangles
Chapter 5
Relationships Within
Triangles
5.5 Inequalities in Triangles
Corollary to the Triangle Exterior Angle Theorem:
The measure of an exterior angle of a triangle
is greater than the measure of each of its
remote interior angles.
2
3
1
m<1 > m<2 and m<1 > m<3
Applying the Corollary
m<2 = m<1 by the Isosceles Triangle Theorem.
Explain why m<2 > m>3.
2
1
4
3
m<1 > m<3 + m<4 because
<1 is the exterior angle, so
m<1 > m<3.
By substitution property, m< 2 > m<3,
since m<2 = m<1.
Theorem 5-10
If two sides of a triangle are not congruent, then
the larger angle lies opposite the longer side.
X
Y
Z 14
12 11
<Y is the largest angle.
Comparing Angles
A landscape architect is designing a triangular
deck. She wants to place benches in the two
larger corners. Which corners have the larger
angles?
21ft
27ft
18ft
A
B
C
<B and <A are the larger angles, <C is the smallest.
Theorem 5-11
If two sides of a triangle are not congruent, then
the longer side lies opposite the larger angle.
X
Y
Z
48
98
34
XZ is the longest side.
Using Theorem 5-11
Which side is the shortest?
52 62
66
U
T
V
TV is the shortest side.
40 60
X
Y
Z
80
YZ is the shortest side
Theorem 5-12
Triangle Inequality Theorem:
The sum of the lengths of any two sides of a
triangle is greater than the length of the third
side.
a
b
c a + b > c
b + c > a
c + a > b
Triangle Inequality Theorem
Can a triangle have sides with the given lengths?
3ft, 7ft, 8ft
3cm, 6cm, 10cm
Yes, 3 + 7 = 10 and 10 > 8
No, 3 + 6 = 9 and 9 is not greater than 10
Triangle Inequality Theorem
Can a triangle have sides with the given lengths?
2m, 7m, 9m
4yd, 6yd, 9yd
No, 2 + 7 = 9, and 9 is not greater than 9
Yes, 4 + 6 = 10 and 10 is greater than 9
Finding Possible Side Lengths
A triangle has side lengths of 8cm and 10cm.
Describe the possible lengths of the third side.
The value of the third side must be greater
Than 2 and less than 18.
(x > 2 and x < 18)
2cm < x < 18cm
To answer this kind of question, add the numbers together and
Subtract the small number from the larger number.
8 + 10 = 18 10 – 8 = 2
Finding Possible Side Lengths
A triangle has side lengths of 3in and 12in.
Describe the possible lengths of the third side.
9in < x < 15in
To answer this kind of question, add the numbers together and
Subtract the small number from the larger number.
3 + 12 = 15 12 – 3 = 9
The value of the third side must be greater
Than 9 and less than 15. (x > 9 and x < 15)
Practice