Objective: 5.3 & Inequalities in One/Two Triangle(s) _________& The Triangle Inequality Warm Up: Solve the inequality: 1. x + 3 < > 10 – x 3. 2x + 3 < 4x – 9 Find the measure of the third angle of a triangle with the two given angle measures °, 59° 5. x°, 2x°

Section 5.3 Inequalities in One Triangle Section 5.5 The Triangle Inequality Section 5.6 Inequalities in Two Triangles

Theorem Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. m  1 > m  A and m  1 > m  B

Theorem 5.9 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. A B C 53 If BC > AB, then m  A > m  C

Ex 1: Comparing Angles A landscape architect is designing a triangular deck. She wants to place benches in the two larger corners. Which corners have the larger angles? 21ft 27ft 18ft A B C m  C < m  A < m  B

Theorem If one angle of triangle is larger than another angle, then the opposite side of the greater angle is longer than the opposite side of the smaller angle. D F E 60  40  If DF

Ex 2: Comparing Sides Which side is the shortest? 52  62  U T V 40  60  X Y Z

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 BC AB + BC > AC AC + BC > AB AB + AC > BC

Ex 3: Can a triangle have sides with the given lengths? 3ft, 2ft, 5ft 4cm, 2cm, 5cm

Ex 4: Finding Possible Side Lengths A triangle has side lengths of 10cm and 14cm. Describe the possible lengths of the third side. Let x represent the length of the third side. Using the Triangle Inequality, write and solve inequalities The length of the third side must be greater than 4cm and less than 24cm.

List the angles in order of Triangle ABC from least to greatest if AB=2x+5, AC=3x-10, BC=x+25 and the perimeter of Triangle ABC is 50.

Hinge Theorem/SAS Inequality If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. RT > VX

Converse of the Hinge Theorem/SSS Inequality If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. m  A > m  D

Ex 3: Finding Possible Side Lengths and Angle Measures Using the Hinge Theorem and its converse, choose possible side lengths or angle measures from a given list. a. AB ≅ DE, BC ≅ EF, AC = 12 inches, m  B = 36°, and m  E = 80°. Which of the following is a possible length for DF? 8 in., 10 in., 12 in., or 23 in.?

Ex 3 cont’d: b.Given ∆RST and ∆XYZ, RT ≅ XZ, ST ≅ YZ, RS = 3.7 cm., XY = 4.5 cm, and m  Z = 75°. Which of the following is a possible measure for  T: 60°, 75°, 90°, or 105°.