5.4 The Triangle Inequality

Objective Apply the Triangle Inequality Theorem

Theorem 5.11 ∆ Inequality Theorem The sum of the lengths of any two sides of a ∆ is greater than the length of the 3rd side. The ∆ Inequality Theorem can be used to determine whether 3 sides can form a triangle or not. d o g d + o > g o + g > d g + d > o

Example 1a: Determine whether the measures and can be lengths of the sides of a triangle. Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle. HINT: If the sum of the two smaller sides is greater than the longest side, then it can form a ∆.

Example 1b: 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.

Your Turn: 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

Example 2: Multiple-Choice Test Item In and Which measure cannot be PR? A 7 B 9 C 11 D 13

Example 2: 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.

Example 2: Graph the inequalities on the same number line. The range of values that fit all three inequalities is

Example 2: Examine the answer choices. The only value that does not satisfy the compound inequality is 13 since 13 is greater than 12.4. Thus, the answer is choice D. Answer: D

Your Turn: Multiple-Choice Test Item Which measure cannot be XZ? A 4 B 9 C 12 D 16 Answer: D

Example 3: Given: line through point J Point K lies on t. Prove: KJ < KH

Example 3: Proof: Statements Reasons 1. 1. Given are right angles. 2. 2. Perpendicular lines form right angles. 3. 3. All right angles are congruent. 4. 4. Definition of congruent angles 5. 5. Exterior Angle Inequality Theorem 6. 6. Substitution 7. 7. If an angle of a triangle is greater than a second angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

Your Turn: Prove: AB > AD Given: is an altitude in ABC.

Your Turn: Proof: Statements 1. 2. 3. 4. Reasons 1. 2. 3. 4. Reasons 1. Given 2. Definition of altitude 3. Perpendicular lines form right angles. 4. All right angles are congruent. is an altitude in are right angles. Continued on next slide 

Your Turn: Proof: Statements 5. 6. 7. 8. Reasons 5. Definition of congruent angles 6. Exterior Angle Inequality Theorem 7. Substitution 8. If an angle of a triangle is greater than a second angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

Assignment Geometry: Pg. 264 #14 – 36, 42 Pre-AP Geometry: Pg. 264 #14 – 37, 42, 44