Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

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Presentation transcript:

Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side

Inequalities in One Triangle Note that there is only one situation that you can have a triangle; when the sum of two sides of the triangle are greater than the third.  They have to be able to reach!!

Triangle Inequality Theorem  AB + BC > AC A B C  AB + AC > BC  AC + BC > AB

Triangle Inequality Theorem A B C  Biggest Side Opposite Biggest Angle  Medium Side Opposite Medium Angle  Smallest Side Opposite Smallest Angle 3 5 m<B is greater than m<C

Triangle Inequality Theorem  Converse is true also  Biggest Angle Opposite _____________  Medium Angle Opposite ______________  Smallest Angle Opposite _______________ B C A Angle A > Angle B > Angle C So CB >AC > AB

Example: List the measures of the sides of the triangle, in order of least to greatest. 10x - 10 = 180 Solving for x: Therefore, BC < AB < AC <A = 2x + 1 <B = 4x <C = 4x -11 2x x + 4x - 11 =180 10x = 190 X = 19 Plugging back into our Angles: <A = 39 o ; <B = 76; <C = 65 Note: Picture is not to scale

Using the Exterior Angle Inequality  Example: Solve the inequality if AB + AC > BC x + 3 x + 2 A B C (x+3) + (x+ 2) > 3x - 2 3x - 22x + 5 > 3x - 2 x < 7

Example: Determine if the following lengths are legs of triangles A)4, 9, ? 9 9 > 9 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: B) 9, 5, 5 Since the sum is not greater than the third side, this is not a triangle ? 9 10 > 9 Since the sum is greater than the third side, this is a triangle

Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third side? 6 12 X = ? = – 6 = 6 Therefore: 6 < X < 18