Section 5.3 Notes: Inequalities in One Triangle EQ: How are the sides related to the angles of a triangle? How are any two sides of a triangle related to the third side?

Definition of Inequality   Definition of Inequality For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles Ex: If 5 = 2 + 3, then 5 > 2 and 5 > 3 Ex: m∠1 > m∠2 and m∠1 > m∠3

Draw an Obtuse Scalene Triangle Find the largest angle and longest side and mark highlight them. Longest side Largest angle

Draw an Obtuse Scalene Triangle Find the smallest angle and shortest side and mark highlight them. What do you notice? Smallest angle Shortest side Largest angle is across from the largest side Smallest angle is across from the shortest side

XY > YZ, so m∠Z > m∠X   Triangle Longer Side Theorem  XY > YZ, so m∠Z > m∠X Triangle Larger Angle Theorem  m∠J > m∠K, so KL > JL Y 9 7 X Z J 110° K 45° L

The largest angle is across from the largest side When given a triangle WITH ONLY SIDE LENGTHS…   How can you use the above theorems to provide the largest angle? The largest angle is across from the largest side When given a triangle with ONLY ANGLE MEASUREMENTS… How can you use the above theorems to provide the longest side? The largest side is across from the largest angle

Example 2 List the angles of ΔABC in order from smallest to largest. m∠C 2) m∠A 3) m∠B

Example 3 List the angles of each triangle in order from smallest to largest as an inequality. m∠C < m∠B < m∠A

Example 5 List the angles of each triangle in order from smallest to largest. ∆XYZ, where XY = 25, YZ = 11, and XZ = 15 Y m∠X < m∠Y < m∠Z 25 11 X Z 15

Example 6 List the sides of each triangle in order from shortest to longest as an inequality. PQ < RQ < PR

Example 7 List the sides of each triangle in order from shortest to longest. UV < UW < WV 70°

Do the you try on your own before you look at the answer

You Try! 1. List the sides of ΔABC in order from shortest to longest. AC < AB < BC 2. List the sides of ΔDEF in order from shortest to longest as equality. DF < FE < DE

B G E D C A F