Triangle Inequalities
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  Theorem 1:

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!

 

Example

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.

 

  Theorem 2:

also ...
 

In a triangle, the longest side is across from the largest angle.
  Theorem 3:

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


Since 7 is the longest side in the triangle, <C, across from it, is the largest angle. 

  


Since 100º is the largest angle in this triangle, , across from it, is 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.


 

Example

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 180º.      80 + 40 + x = 180
           120 + x = 180
                     x = 60
 

We have now found that <B measures 60º.  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.

 


 

  Theorem 4:

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.



 

Example
Suppose we are faced with the following proof:

    

Statements Reasons
1. 1. Given
2. 2. Congruent angles are angles of equal measure.
3. 3.

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