The Triangle Inequality

1
Lesson 5-4

• The Triangle Inequality

2
Transparency 5-4
5-Minute Check on Lesson 5-3
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 ? 3. 5.
?MNO is an equilateral triangle. 6.
Which statement is a
contradiction to the statement that ?W and ?V are
vertical angles?
Standardized Test Practice
m?W 85?
B
A
?W ? ?V
D
m?W gt m?V
?W is acute
C
3
Transparency 5-4
5-Minute Check on Lesson 5-3
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 gt 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
m?W 85?
B
A
?W ? ?V
D
m?W gt m?V
?W is acute
C
4
Objectives

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

5
Vocabulary

• No new vocabulary words or symbols

6
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.

7
Triangle Inequality
Can a triangle be made out of these pieces?
7.5
4.4
3.3
Yes
The sum of any two sides is greater than the
third.
8
Answer Because the sum of two measures is not
greater than the length of the third side, the
sides cannot form a triangle.
9
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.
10
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
11
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 lt RP lt RQ QP In numbers 14
10 lt RP lt 10 14 4 lt RP lt 24
12
A 7 B 9 C 11 D 13
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.
13
Graph the inequalities on the same number line.
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
14
A 3 B 9 C 12 D 14
Answer D
15
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