Lesson 5-4 The Triangle Inequality 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.

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. Example 4-1a

Determine whether the measures 6. 8, 7. 2, and 5 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. Example 4-1b

Determine whether the given measures can be lengths of the sides of a triangle. Answer: no Answer: yes Example 4-1c

Multiple-Choice Test Item In and Which measure cannot be PR? A 7 B 9 C 11 D 13 Example 4-2a

You need to determine which value is not valid. 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 4-2a

Graph the inequalities on the same number line. The range of values that fit all three inequalities is Example 4-2a

Examine the answer choices 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 Example 4-2a

Multiple-Choice Test Item Which measure cannot be XZ? A 4 B 9 C 12 D 16 Answer: D Example 4-2b

Given: line through point J Point K lies on t. Prove: KJ < KH Example 4-3a

2. Perpendicular lines form right angles. 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. Example 4-3a

Given: is an altitude in ABC. Prove: AB > AD Given: is an altitude in ABC. Example 4-3b

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. Example 4-3b

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. Example 4-3b