Sect. 5.5 Inequalities in One Triangle Goal 1 Comparing Measurements of a Triangle. Goal 2 Using the Triangle Inequality.

Comparing Measurements of a Triangle Theorem 5.10 If one side of a triangle is longer than a second side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. Theorem 5.10 The largest angle in  ABC is  A.

Comparing Measurements of a Triangle Theorem 5.11 If one angle of a triangle has a greater measure than a second angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. The longest side in  RST is

Comparing Measurements of a Triangle Write the measures of the sides of the triangle in order from least to greatest.

Comparing Measurements of a Triangle Write the measures of the angles of the triangle in order from least to greatest.

Comparing Measurements of a Triangle a)Name the smallest and largest angles of  PQR. b) Is QR ≥ 8? Why? c) Is PQ < 8? Why?

Comparing Measurements of a Triangle Recall! Exterior Angle – When sides of a triangle are extended, exterior angles are adjacent and supplementary to interior angles.  1 is an exterior angle Exterior Angle Theorem An Exterior Angle is equal to the sum of the two remote interior angles.  1 =  3 +  4

If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. Theorem 5.12 Exterior Angle Inequality Comparing Measurements of a Triangle

Using the Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Theorem 5-12 Triangle Inequality Theorem AB + BC > AC AC + BC > AB AB + AC > BC

Using the Triangle Inequality The Triangle Inequality basically says that you need long enough sides so that they reach each other. It is showing the growth of two segments until they meet. Before they meet no triangle is formed

Using the Triangle Inequality Hint - For example would sides of length 4, 5 and 6 form a triangle....? How about sides of length 4, 11, and 7? If you are an observant student, then you noticed that all you have to do is add the two smallest sides to see if it is larger than the other!

Using the Triangle Inequality Arrange the sides of quadrilateral ABCD in order from smallest to largest.

Using the Triangle Inequality Which of the following sets of numbers could represent the lengths of the sides of a triangle? a) 3, 4, 6b) 10, 11, 21c) 2, 6, 9d) 34, 35, 36

Using the Triangle Inequality A triangle has one side of 11 inches and another side of 16 inches. a) Describe the possible lengths of the 3 rd side. b) Construct a possible triangle with the two given sides.

Homework even, 42-46