Triangle Inequalities MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts

 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 will not have a triangle!

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

 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.
 Since 9 is the longest side of the triangle, ∠C (across from it) is the largest angle. Since 88º is the largest angle of the triangle, (across from it) is the longest side.

 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.

 Examples:

 1 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 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. is the longest side.
 3. 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

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