Honors Geometry

Relationships in Triangles Chapter 5 Relationships in Triangles

The Triangle Inequality Section 5-5  The Triangle Inequality Objective: To use the Triangle Inequality Theorem to identify possible triangles To prove triangles relationships using the Triangle Inequality Theorem.

Theorem 5.11: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Given: ΔABC Prove: (1) AB + BC > AC (2) AB + AC > BC (3) AC + BC > AB C A B The Triangle Inequality Theorem can be used to determine whether three segments can form a triangle.

Example: Is it possible for a triangle to have sides with the lengths indicated below. 1.) 10, 9, 8 Yes 2.) 6, 6, 20 No 3.) 7, 7, 14.1 No 4.) 16, 11, 5 No 5.) 0.6, 0.5, 1 Yes 6.) 18, 18, 0.06 Yes

Example: Find the range for the measure of the third side of a triangle given the measures of two sides. 5, 7, n

n < 12 n > 2 n > -2 2 < n < 12 6 5 3 4 2 9 7 8 10 1 11 -1 6 5 3 4 2 9 7 8 10 1 11 12 -2 n > 2 -1 6 5 3 4 2 9 7 8 10 1 11 12 -2 n > -2 -1 6 5 3 4 2 9 7 8 10 1 11 12 -2 -1 2 < n < 12

Example: Find the range for the measure of the third side of a triangle given the measures of two sides. 5, 7, n 2 < n < 12

2 < n < 10 8 < n < 14 0 < n < 24 0 < n < 10 Example: Find the range for the measure of the third side of a triangle given the measures of two sides. 1.) 4 and 6 2 < n < 10 2.) 3 and 11 8 < n < 14 3.) 12 and 12 0 < n < 24 4.) 5 and 5 0 < n < 10

End of Section 5.5