Geometry
Lesson 6.4
Inequalities
for One Triangle
Tomorrow we will learn inequalities
For Two Triangles
The Main Idea
You can think of an angle as the jaw bones
of an alligator.
When you prop the jaw open with a stick,
you form a triangle.
The larger the stick,
the larger the angle
formed by the
jawbones.
The smaller the stick,
the smaller the angle
formed by the
jawbones.
You could prop a very long
stick in the mouth of a
hippo, creating a very large
angle.
long
stick
large
angle
It follows logically that the
longest side of a triangle
would be opposite the largest
angle
And the shortest side would
be opposite the smallest
angle.
Theorem
If one side of a triangle is longer than a second
side, then the angle opposite the first side is
larger than the angle opposite the second
side.
A
If BC > AB, then
small
m<A > m<C.
C
big
B
Crikey,
that seems easy!
Longest side
opposite biggest
angle, shortest side
opposite smallest
angle, medium side
opposite medium
angle.
Examples: What ’s are the largest and smallest?
A
2.
A
1.
x -3
x + 10
10
6
B
B
C
9
Largest : _____; Smallest : ______
C
x
Largest : _____; Smallest : ______
Examples: What sides are the largest and smallest?
A
3.
A
4.
5x
90
B
60
30
C
Largest side: _____; Smallest side:_____
B
2x
10x
C
Largest side: _____; Smallest side:_____
Consider two sides of a triangle, each
side 6 inches long
6
6
What is the maximum length the third side
can be?
3
5
9
6
6
6
6
3
5
6
6
11.999
6
6
12
6
6
9
How long could the
third side be?
Could it be 12 or
more inches long?
No – ∆ could NOT be drawn
Theorem
The sum of the lengths of any two sides of a
triangle is greater than the length of the
third side.
9 + 11 = 20
20 > 14
A
11 + 14 = 25 25 > 9
14 + 9 = 23
14
23 > 11
C
9
11
B
Example
The lengths of two sides of a triangle are 3 and 5.
The length of the third side must be greater than
________ and less than _______.
Let x be the length of the third side.
x+3>5
x>
2
3+5>x
8>x
x<
8
x+5>3
x > -2
Examples: Is this a triangle with these side measures possible?
5.
10, 9, 8
6.
16, 11, 5
Sample Problems
1.
2.
A
x
B
3. ABCD is parallelogram
A
A
x-5
x+3
D
x
C
x-3
54
52
C
B
B
Largest : ____;
Smallest : ____
Largest : ____;
Smallest : ____
4.
5.
C
53
Largest : ____ _____;
Smallest : ________
6.
A
A
B
A
68
60
60
B
D 3x
59
B
54
Largest side: _________
Smallest side: _______
11x
C
58
C
D
Largest side: ________
Smallest side: ________
4x
C
Longest Side: ____
Given the possible sides listed, do we
have a triangle:
The 3rd side of this triangle must be
between what two numbers:
The 3rd side of this triangle must be
between what two numbers:
7.
9.
11.
8.
5, 11, 15
10, 10, 0.5
Sides: 5, 15
Sides: 4, 5
_____ < 3rd side <______
_____ < 3rd side <______
10.
Sides: 20, 14
12.
_____ <
3rd
side <_____
Sides: 3x, 7x
_____ < 3rd side <______
Homework
pg. 222 #1-17
Corollaries
The segment from a point to a line is the
shortest segment from the point to the line.
●
The segment from a point to a plane is the
shortest segment from the point to the plane.
●
The following is an
Anti- Drug Message…….
Examples
A
Which is the largest angle?
6
10
Which is the smallest angle?
B
<B
C
8
A
20˚
120˚ B
40˚
C
Name the side lengths from
shortest to longest.
BC
AB
AC
<C
Nude crack man in alligator rescue
Thursday, November 30, 2006
In further news of the misadventures of people who are naked, a
pulled from the jaws of an alligator in Florida on Wednesday.
naked man on crack was
He was only saved when sheriff's deputies jumped into the lake where he was being eaten, and
wrestled with the alligator until it let go of the man, before pulling the man about 40m back to the shore.
They had responded to several reports from members of the public that screams had been heard
coming from the lake, at around 4 o'clock in the morning.
The man – identified as 45-year-old Adrian Apgar – lost his left arm in the alligator attack, as
well as suffering severe injuries to his right arm and left leg. His right leg is believed to be fine.
'He was totally naked,' noted Sheriff Grady Judd.
He admitted to the deputies after he was pulled from the water that he'd been smoking crack. No
explanation has yet been offered for why he was naked, although police have not ruled out the
possibility that he might have decided to go for a swim.
Judd added: 'He admitted that he'd been smoking crack cocaine. But still, it's a human life.
Our deputies don't ask questions, they respond and they save people.'
At the time of writing, the condition of the alligator is unknown.
ty Images
In Memory
Steve Irwin
1962 - 2006