
The
sum of the lengths of any two sides of a triangle must be
greater than the
third side. 




If these inequalities are NOT true, you do
not have a triangle!


Suppose
we know the lengths of two sides of a triangle, and we want to
find the "possible" lengths of the third side. 
According
to our theorem, the following 3 statements must be true: 

5 + x
> 9
So,
x
> 4 
5 + 9
> x
So,
14 > x 
x + 9
> 5
So,
x
> 4
(no
real information is gained here since the lengths of the sides must
be positive.) 
Putting
these statements together, we get that
x
must be
greater
than 4,
but
less
than 14.
So any number in the range
4 <
x
< 14 can represent the length of the missing side of our triangle. 

In
a triangle,
the longest side is across from the largest angle. 


In a triangle,
the largest angle is across from the longest side. 



These theorems can be modified to apply to a discussion of only
two angles within the triangle:
Theorem: In a triangle, the
longer side is across from the larger angle.
Theorem: In a triangle, the
larger angle is across from the longer side.
Suppose
we want to know which side of this triangle is the longest. 

Before
we can utilize our theorem, we need to know the size of <B. We
know that the 3 angles of the triangle add up to 180°. 
80
+ 40 + x = 180
120 + x = 180
x = 60 
We have now found that <B measures 60°.
According to our theorem, the longest side will be across from the
largest angle. 
Now
that we know the measures of all 3 angles, we can tell that <A is
the largest. This means the side across from <A,
side , is
the longest side. 

The measure of
the exterior angle of a triangle is greater than the measure of
either nonadjacent interior angle. 



<1 is the exterior angle.
<2 and <3 are its nonadjacent interior angles. 

Suppose
we are faced with the following proof: 


