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: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a + b > c a + c > b b + c > a 4

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

Triangle Inequality Theorem: Can you make a triangle? 6

Theorem If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle. smaller angle larger angle longer side shorter side A B C 7

In a Triangle: 4 The largest angle is opposite the largest side. 4 The smallest angle is opposite the smallest side. 4 The smallest side is opposite the smallest angle. 4 The largest side is opposite the largest angle. 8

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

Finding the range of the third side : Given The lengths of two sides of a triangle Since the third side cannot be larger than the other two added together, we find the maximum value by adding the two sides. Since the third side and the smallest side given cannot be larger than the other side, we find the minimum value by subtracting the two sides. Difference < Third Side < Sum 12

Finding the range of the third side : Example Given a triangle with sides of length 3 and 7, find the range of possible values for the third side. Solution Let x be the length of the third side of the triangle. The maximum value: x < = 10 The minimum value: x > 7 – 3 = 4 So 4 < x < 10 (x is between 4 and 10.) x x x < 10 x > 4 13

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