Transcript 5.5 Inequalities in Triangles

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.
3
2
m<1 > m<2
1
and
m<1 > m<3
Applying the Corollary
m<2 = m<1 by the Isosceles Triangle Theorem.
Explain why m<2 > m>3.
3
1
m<1 > m<3 + m<4 because
<1 is the exterior angle, so
m<1 > m<3.
4
By substitution property, m< 2 > m<3,
since m<2 = m<1.
2
Theorem 5-10
If two sides of a triangle are not congruent, then
the larger angle lies opposite the longer side.
Y
11
12
X
14
<Y is the largest angle.
Z
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?
A
27ft
C
18ft
21ft
B
<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.
Y
98
48
34
X
Z
XZ is the longest side.
Using Theorem 5-11
Which side is the shortest?
T
66
U
52
TV is the shortest side.
62
V
Y
80
YZ is the shortest side
60
X
40
Z
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.
c
a
a+b>c
b+c>a
c+a>b
b
Triangle Inequality Theorem
Can a triangle have sides with the given lengths?
3ft, 7ft, 8ft
Yes, 3 + 7 = 10 and 10 > 8
3cm, 6cm, 10cm
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
No, 2 + 7 = 9, and 9 is not greater than 9
4yd, 6yd, 9yd
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.
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
The value of the third side must be greater
Than 2 and less than 18.
(x > 2 and x < 18)
2cm < x < 18cm
Finding Possible Side Lengths
A triangle has side lengths of 3in and 12in.
Describe the possible lengths of the third side.
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)
9in < x < 15in
Practice
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