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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.
d + o > g
o + g > d
g + d > o

o
d
g
The ∆ Inequality Theorem can be used to
determine whether 3 sides can form a
triangle or not.
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
Answer: no
b. 14, 16, 27
Answer: yes
Example 2:
Multiple-Choice Test Item
In
and
be PR?
A 7
B9
C 11
Which measure cannot
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
Answer: D
B9
C 12
D 16
Example 3:
Given:
Prove:
line
through point J
Point K lies on t.
KJ < KH
Example 3:
Proof:
Statements
1.
2.
are right angles.
3.
4.
5.
6.
7.
Reasons
1. Given
2. Perpendicular lines form right
angles.
3. All right angles are congruent.
4. Definition of congruent angles
5. Exterior Angle Inequality Theorem
6. Substitution
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:
Given:
is an altitude in ABC.
Prove: AB > AD
Your Turn:
Proof:
Statements
is an altitude
1.
Reasons
1. Given
in
2.
3.
are right angles.
4.
2. Definition of altitude
3. Perpendicular lines form
right angles.
4. All right angles are congruent.
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