Lesson 5-4 The Triangle Inequality

5-Minute Check on Lesson 5-3 Transparency 5-4 Write the assumption you would make to start an indirect proof of each statement. 1.  ABC   DEF 2. RS is an angle bisector. 3.  X is a right angle. 4. If 4x – 3  9, then x   MNO is an equilateral triangle. 6. Which statement is a contradiction to the statement that  W and  V are vertical angles? Standardized Test Practice: A C B D m  W > m  V W  VW  V m  W = 85   W is acute

5-Minute Check on Lesson 5-3 Transparency 5-4 Write the assumption you would make to start an indirect proof of each statement. 1.  ABC   DEF  ABC ≇  DEF 2. RS is an angle bisector.RS is not an angle bisector. 3.  X is a right angle.  X is not a right angle. 4. If 4x – 3  9, then x  3.x > 3 5.  MNO is an equilateral triangle.  MNO is not an equilateral triangle. 6. Which statement is a contradiction to the statement that  W and  V are vertical angles? Standardized Test Practice: A C B D m  W > m  V W  VW  V m  W = 85   W is acute

Objectives Apply the Triangle Inequality Theorem Determine the shortest distance between a point and a line

Vocabulary No new vocabulary words or symbols

Theorems & Corollaries Theorem 5.11, Triangle Inequality Theorem – The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Theorem 5.12 – The perpendicular segment from a point to a line is the shortest segment from the point to the line. Corollary 5.1 – The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

Triangle Inequality Can a triangle be made out of these pieces? Yes The sum of any two sides is greater than the third.

Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle. Determine whether the measures and can be lengths of the sides of a triangle.

Determine whether the measures 6.8, 7.2, and 5.1 can be lengths of the sides of a triangle. Check each inequality. Answer: All of the inequalities are true, so 6.8, 7.2, and 5.1 can be the lengths of the sides of a triangle.

Determine whether the given measures can be lengths of the sides of a triangle. a. 6, 9, 16 b. 14, 16, 27 Answer: no Answer: yes

Triangle Inequality Revisited Given two sides of a triangle, what can the third be? In a triangle PQR with RQ = 10 and QP = 14, what can RP be? Any two sides must be greater than the third, so QP – RQ < RP < RQ + QP In numbers 14 – 10 < RP < < RP < 24

A 7 B 9 C 11 D 13 Multiple-Choice Test Item In and Which measure cannot be PR? Read the Test Item You need to determine which value is not valid. Solve the Test Item Solve each inequality to determine the range of values for PR.

Graph the inequalities on the same number line. The range of values that fit all three inequalities is Examine the answer choices. The only value that does not satisfy the compound inequality is 13 since 13 is greater than Thus, the answer is choice D. Answer: D

A 3B 9C 12D 14 Answer: D Multiple-Choice Test Item Which measure cannot be XZ?

Summary & Homework Summary: –The sum of the lengths of any two sides of a triangle is greater then the length of the third side. Homework: –pg 264: 15-19, 27-31