1/18 Warm Up
Use the following diagram for numbers 1 – 2.
The perpendicular bisectors of ∆ABC meet at D.
1. Find DB.
2. Find AE.
A
G
22
14
E
D
B
C
F
B
Use the following diagram for numbers 6.
The angle bisectors of ∆ABC meet at P.
3. Find x.
31
Use the following diagram and answer choices for #10-15
is the angle bisector of EFG
FH
4. FG = _______
E
.
10 cm
F
H
32°
6. m EFG = _______
7. EH = _______
8. m  FGH = ________
9. m GFH = _______ 10. m FEH = ________
5 cm
G
January 20, 2016
Geometry 5.5 Inequalities in One Triangle
1
Geometry
6.5 Inequalities in One Triangle
[email protected]
6.5 & 6.6 Essential Question
How are the sides related to the angles of a
triangle?
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Geometry 5.1 Perpendiculars and Bisectors
3
Goals



Use the Triangle Inequality Theorem.
Be able to determine the largest and smallest
angles and sides of a triangle.
Use the Hinge Theorem and its converse to
compare sides lengths and angle measures
of two triangles.
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Geometry 5.5 Inequalities in One Triangle
4
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Geometry 5.5 Inequalities in One Triangle
5
Can these three sides
form a triangle?
2
2
5
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Geometry 5.5 Inequalities in One Triangle
6
Can these three sides
form a triangle?
2
1
2
5
No: 2 + 2 leaves a gap of 1 in the middle.
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Geometry 5.5 Inequalities in One Triangle
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Geometry 5.5 Inequalities in One Triangle
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Can these three sides
form a triangle?
2
2
5
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Geometry 5.5 Inequalities in One Triangle
9
2
2
5
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Geometry 5.5 Inequalities in One Triangle
10
2
2
5
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2
2
5
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Geometry 5.5 Inequalities in One Triangle
12
2
2
5
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Geometry 5.5 Inequalities in One Triangle
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2
2
5
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Geometry 5.5 Inequalities in One Triangle
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2
2
5
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Geometry 5.5 Inequalities in One Triangle
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2
2
5
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Geometry 5.5 Inequalities in One Triangle
16
2
2
5
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Geometry 5.5 Inequalities in One Triangle
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2
2
5
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2
2
5
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Geometry 5.5 Inequalities in One Triangle
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No. Sides of 2 & 2 leave a gap
of length 1.
(Notice: 2 + 2 < 5)
2
1
2
5
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Geometry 5.5 Inequalities in One Triangle
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Geometry 5.5 Inequalities in One Triangle
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Can these three sides
form a triangle?
3
2
5
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Geometry 5.5 Inequalities in One Triangle
22
Can these three sides
form a triangle?
3
2
5
No: 3 + 2 exactly matches the 5
leaving no room for an angle.
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Geometry 5.5 Inequalities in One Triangle
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Geometry 5.5 Inequalities in One Triangle
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Can these three sides
form a triangle?
3
2
5
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Geometry 5.5 Inequalities in One Triangle
25
3
2
5
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Geometry 5.5 Inequalities in One Triangle
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3
2
5
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3
2
5
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3
2
5
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3
2
5
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3
2
5
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2
3
5
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3
2
5
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3
2
5
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3
2
5
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Geometry 5.5 Inequalities in One Triangle
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3
2
5
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3
2
5
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3
2
5
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3
2
5
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3
2
5
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3
2
5
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3
2
5
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No. Lengths of 3 & 2 fit the side
of length 5 exactly.
(Notice: 2 + 3 = 5)
3
2
5
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Will these lengths
form a triangle?
4
3
5
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Geometry 5.5 Inequalities in One Triangle
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4
3
5
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4
3
5
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4
3
5
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4
3
5
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4
3
5
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4
3
5
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4
3
5
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4
3
5
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4
3
5
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Yes!
4
3
5
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Geometry 5.5 Inequalities in One Triangle
55
Together, lengths of 3 & 4 are longer
than 5 and will meet before collapsing
all the way.
(Notice: 3 + 4 > 5)
4
3
5
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Geometry 5.5 Inequalities in One Triangle
56
Triangle Inequality Theorem (6.11)

In any triangle, the sum of any two sides is
greater than the third side.
a+b>c
b+c>a
a+c>b
All of these
must be true.
January 20, 2016
b
a
Geometry 5.5 Inequalities in One Triangle
c
57
Example 1
Is this triangle possible?
Yes
 4 + 5 > 7
 7 + 4 > 5
 5 + 7 > 4
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5
4
7
Geometry 5.5 Inequalities in One Triangle
58
Example 2
Is this triangle possible?
NO
 12 + 4 > 7
 12 + 7 > 4
 But,
 7 + 4 > 12
7
12
7
January 20, 2016
4
12
1
Geometry 5.5 Inequalities in One Triangle
4
59
Your Turn
Is this triangle possible?
Yes
 2 + 9 > 10
 10 + 2 > 9
 2 + 10 > 9
January 20, 2016
9
2
10
Geometry 5.5 Inequalities in One Triangle
60
Given two sides, what is the range of the
third side of a triangle?
a=3
b=5
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Geometry 5.5 Inequalities in One Triangle
61
Given two sides, what is the range of the
third side of a triangle?
Side c would be 2: just
enough to cover side b.
No Triangle.
c=2
a=3
b=5
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Geometry 5.5 Inequalities in One Triangle
62
Rotate side a…
a=3
c=2
b=5
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Geometry 5.5 Inequalities in One Triangle
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Rotate side a…
a=3
c=2
b=5
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Geometry 5.5 Inequalities in One Triangle
64
Rotate side a…
a=3
c increases
b=5
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Geometry 5.5 Inequalities in One Triangle
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Rotate side a…
a=3
c increases
b=5
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Rotate side a…
a=3
c increases
b=5
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Geometry 5.5 Inequalities in One Triangle
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Rotate side a…
c increases
a=3
b=5
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Geometry 5.5 Inequalities in One Triangle
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Rotate side a…
c increases
a=3
b=5
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Geometry 5.5 Inequalities in One Triangle
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Rotate side a…
a=3
c increases
b=5
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70
Rotate side a…
a=3
c increases
b=5
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Geometry 5.5 Inequalities in One Triangle
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Rotate side a…
c increases
a=3
b=5
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Geometry 5.5 Inequalities in One Triangle
72
Rotate side a…
c increases
a=3
b=5
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Geometry 5.5 Inequalities in One Triangle
73
Rotate side a…
c increases
a=3
b=5
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Geometry 5.5 Inequalities in One Triangle
74
Rotate side a…
c increases
a=3
b=5
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Geometry 5.5 Inequalities in One Triangle
75
Rotate side a…
c increases
a=3
b=5
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Rotate side a…
c increases
a=3
b=5
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Rotate side a…
c increases
a=3
b=5
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Rotate side a…
c increases
a=3
b=5
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Rotate side a…
c increases
a=3
b=5
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80
Rotate side a…
c increases
a=3
b=5
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81
Rotate side a…
c increases
a=3
b=5
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82
Rotate side a…
c increases
a=3
b=5
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83
Rotate side a…
c increases
a=3
b=5
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84
Rotate side a…
c increases
a=3
b=5
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85
Rotate side a…
c increases
a=3
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b=5
Geometry 5.5 Inequalities in One Triangle
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Rotate side a…
c increases
a=3
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b=5
Geometry 5.5 Inequalities in One Triangle
87
Rotate side a…
c increases
a=3
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b=5
Geometry 5.5 Inequalities in One Triangle
88
Rotate side a…
c=?
a=3
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b=5
Geometry 5.5 Inequalities in One Triangle
89
No triangle – sides fall
on top of each other.
c=8
a=3
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b=5
Geometry 5.5 Inequalities in One Triangle
90
The extremes of c…
a=3
c=2
b=5
c=8
a=3
January 20, 2016
b=5
Geometry 5.5 Inequalities in One Triangle
91
Corollary to Triangle Inequality Theorem

If two sides of a triangle measure a and b, with
a being the larger side, then the third side, c, is
greater than a – b and less than a + b.
a-b < c < a+b
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Geometry 5.5 Inequalities in One Triangle
92
Example 3
c
8
12
c is greater than ________
12 – 8 and less than
_________.
12 + 8
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Geometry 5.5 Inequalities in One Triangle
93
Example 3
c
8
12
c is greater than 4 and less than 20.
Or, c is between 4 and 20.
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Geometry 5.5 Inequalities in One Triangle
94
Your Turn
What are the possible values of x?
75
10
x
x is between 65 and 85.
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Geometry 5.5 Inequalities in One Triangle
95
Two Other Theorems

6.9: If one side of a triangle is longer than
another side, then the angle opposite the
larger side is larger than the angle opposite the
shorter side. (Given the side lengths, the
largest angle is opposite the longest side.)

6.10: If one angle of a triangle is larger than
another angle, then the side opposite the
larger angle is longer than the side opposite
the smaller angle. (Given the angle measures,
the longest side is opposite the largest angle.)
January 20, 2016
Geometry 5.5 Inequalities in One Triangle
96
Example 4




List the angles of the triangle in order from
smallest to largest.
S
T
14
R
8
S
R
T
15
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Geometry 5.5 Inequalities in One Triangle
97
Example 5




List the sides of the triangle in order from
smallest to largest.
b
70
c
b
a
a
80
30
c
January 20, 2016
Geometry 5.5 Inequalities in One Triangle
98
The Hinge Theorem
c
a
d
a
2
1
b
b
Begin with two congruent triangles.
(SAS)
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
99
The Hinge Theorem
c
a
d
a
2
1
b
b
Rotate 1 to make it smaller.
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
100
The Hinge Theorem
c
a
d
a
2
1
b
b
Rotate 1 to make it smaller.
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Geometry 5.6 Inequalities in Two Triangles
101
The Hinge Theorem
c
a
d
a
2
1
b
b
Rotate 1 to make it smaller.
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Geometry 5.6 Inequalities in Two Triangles
102
The Hinge Theorem
c
a
d
a
2
1
b
b
What happens to side c?
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103
The Hinge Theorem
c
a
d
a
2
1
b
b
What happens to side c?
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Geometry 5.6 Inequalities in Two Triangles
104
The Hinge Theorem
a
c
d
a
2
1
b
b
What happens to side c?
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Geometry 5.6 Inequalities in Two Triangles
105
The Hinge Theorem
a
c
d
a
2
1
b
b
What happens to side c?
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Geometry 5.6 Inequalities in Two Triangles
106
The Hinge Theorem
a
c
d
a
2
1
b
b
What happens to side c?
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Geometry 5.6 Inequalities in Two Triangles
107
The Hinge Theorem
a
c
d
a
2
1
b
b
What happens to side c?
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Geometry 5.6 Inequalities in Two Triangles
108
The Hinge Theorem
a
c
d
a
2
1
b
b
What happens to side c?
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Geometry 5.6 Inequalities in Two Triangles
109
The Hinge Theorem
a
c
d
a
2
1
b
b
What happens to side c?
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
110
The Hinge Theorem
a
c
d
a
2
1
b
b
It gets smaller.
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
111
The Hinge Theorem
a
c
d
a
2
1
b
b
Therefore, if 1 is smaller than 2,
then side c is smaller than side d.
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
112
Hinge Theorem
If two sides of one triangle are congruent to two
sides of another triangle, and the included angle of
the first is larger than the included angle of the
second, then the third side of the first is longer
than the third side of the second.
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
113
Converse of the Hinge Theorem
If two sides of one triangle are congruent to two
sides of another triangle, and the third side of the
first is longer than the third side of the second,
then the included angle of the first is larger than
the included angle of the second.
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
114
An Easy Memory Aid


The smaller the angle, the smaller the
opposite side.
The larger the angle, the larger the opposite
side.
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
115
Think of a door.
As the angle at the hinge increases,
the size of the opening increases.
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
116
Example 6
How does x compare to y?
5
x
y
5
30
6
6
Since 30 < 90, x < y.
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Geometry 5.6 Inequalities in Two Triangles
117
Example 7
How does mA compare to mR?
C
T
16
10
A
15
17
10
B
R
15
S
Since 16 < 17, mA < mR.
January 20, 2016
Geometry 5.6 Inequalities in Two Triangles
118
Summary




In a triangle, the sum of any two sides is
greater than the third side.
If two sides of a triangle measure a and b, then
side c is between a – b and a + b.
In any triangle, the largest side is opposite the
largest angle and the smallest side is opposite
the smallest angle.
Given two triangles with two congruent sides,
the longer side is opposite the larger angle and
the larger angle is opposite the longer side.
January 20, 2016
Geometry 5.5 Inequalities in One Triangle
119
Homework
6.5 & 6.6 worksheet
January 20, 2016
Geometry 5.5 Inequalities in One Triangle
120
The sum of any two sides of a triangle is
greater than the remaining side.
Will these three segments form a triangle?
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Geometry 5.5 Inequalities in One Triangle
121