theorem1
The sum of the lengths of any two sides of a triangle must be greater than the third side.
hintgal
If these inequalities are NOT true, you will not have a triangle!

AB + AC > CB    ( 9 + 7 > 5)
AC + CB > AB
  (7 + 5 > 9)
CB + AB > AC
  (5 + 9 > 7)

ineq1


theorem1
Converse:
In a triangle, the longest side is across from the largest angle.
In a triangle, the largest angle is across from the longest side.
Both of these theorems may also be stated using "longer" and " larger" when dealing with 2 sides and 2 angles.
ineq1
Since 9 is the longest side of the triangle,
C (across from it) is the largest angle.
 
ineq3
Since 88º is the largest angle of the triangle,
rs (across from it) is the longest side.


theorem1
The measure of the exterior angle of a triangle is greater than the measure of either non-adjacent interior angle.

This is one of those "common sense" theorems.
In the diagram at the right, ∠1 is an exterior angle for ΔABC.
By the Exterior Angle Theorem, m∠1 = m∠2 + m∠3.
It is common sense that m∠1 > m∠ 2 and m∠1 > m∠3.

ineq6

Examples:

1.
INEQ4
Given the 2 sides shown,
find the "possible" lengths
of the third side.

Solution:
• 8 + x > 12, so x > 4
• x + 12 > 8, so x > -4
(no info, length positive)
• 8 + 12 > x, so 20 > x
Putting the statements together, we have x must be
greater than 4, but less than 20.
4 < x < 20

2.
ineq5
Given the 2 angles shown,
determine which side is the
"longest" side of the triangle.
Solution:
We must find m∠B to determine if it is larger than 62º, making it the largest angle in the triangle.
m∠A + m∠B + m∠C = 180º
62º + m∠B + 55º = 180º
m∠B = 63º, making ∠B the largest angle in the triangle.
ac is the
longest side.
3.
ineq7

Solution:
1) Exterior Angle Theorem - TRUE
2) Inequality Theorem about Exterior Angles
(stated above) - TRUE
3) Linear Pairs are supplementary (2 ∠s adding to 180) - TRUE
4)
FALSE (it should read m∠1 > m∠C)

 
Given ΔABC as shown.
Which statement is NOT true?
1) m∠1 = m∠A + m∠C
2) m∠1 > m∠A
3) m∠1 + m∠ABC = 180º
4) m∠1 < m∠C


divider

NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use".
A