The Triangle Inequality

1
The Triangle Inequality Inequalities in
Multiple Triangles
2
Objectives

• Apply the Triangle Inequality Theorem
• Recognize and apply properties of inequalities to
  the measures of angles in a triangle
• Recognize and apply properties of inequalities to
  the relationships between angles and sides of
  triangles

3
Inequalities

• An inequality simply shows a relationship between
  any real numbers a and b such that if a gt b then
  there is a positive number c so a b c.
• All of the algebraic properties for real numbers
  can be applied to inequalities and measures of
  angles and segments (i.e. multiplication,
  division, and transitive).

4
Example 1
Determine which angle has the greatest measure.
Explore Compare the measure of ?1 to the measures
of ?2, ?3, ?4, and ?5.
Plan Use properties and theorems of real numbers
to compare the angle measures.
5
Example 1
Solve Compare m?3 to m?1.
Compare m?4 to m?1.
Compare m?5 to m?1.
6
Example 1
Compare m?2 to m?5.
Examine The results on the previous slides show
that m?1 gt m?2, m?1 gt m?3, m?1 gt m?4, and m?1 gt
m?5. Therefore, ?1 has the greatest measure.
Answer ?1 has the greatest measure.
7
Your Turn
Determine which angle has the greatest measure.
Answer ?5 has the greatest measure.
8
Exterior Angle Inequality Theorem

• If an ? is an exterior ? of a ?, then its measure
  is greater than the measure of either of its
  remote interior ?s.

m ?1 gt m ?3m ?1 gt m ?4
9
Example 2a
Use the Exterior Angle Inequality Theorem to list
all angles whose measures are less than m?14.
By the Exterior Angle Inequality Theorem, m?14 gt
m?4, m?14 gt m?11, m?14 gt m?2, and m?14 gt m?4
m?3. Since ?11 and ?9 are vertical angles, they
have equal measure, so m?14 gt m?9. m?9 gt m?6 and
m?9 gt m?7, so m?14 gt m?6 and m?14 gt m?7. Answer
Thus, the measures of ?4, ?11, ?9, ? 3, ? 2, ?6,
and ?7 are all less than m?14 .
10
Example 2b
Use the Exterior Angle Inequality Theorem to list
all angles whose measures are greater than m?5.
By the Exterior Angle Inequality Theorem, m?10 gt
m?5, and m?16 gt m?10, so m?16 gt m?5, m?17 gt m?5
m?6, m?15 gt m?12, and m?12 gt m?5, so m?15 gt m?5.
Answer Thus, the measures of ?10, ?16, ?12, ?15
and ?17 are all greater than
m?5.
11
Your Turn
Answer ?5, ?2, ?8, ?7
Answer ?4, ?9, ?5
12
Theorem 5.9

• If one side of a ? is longer than another side,
  then the ? opposite the longer side has a greater
  measure then the ? opposite the shorter side
  (i.e. the longest side is opposite the largest
  ?.)

2
m ?1 gt m ?2 gt m ?3
3
1
13
Example 3a
Determine the relationship between the measures
of ?RSU and ?SUR.
Answer The side opposite ?RSU is longer than
the side opposite ?SUR, so m?RSU gt m?SUR.
14
Example 3b
Determine the relationship between the measures
of ?TSV and ?STV.
Answer The side opposite ?TSV is shorter than
the side opposite ?STV, so m?TSV lt m?STV.
15
Example 3c
Determine the relationship between the measures
of ?RSV and ?RUV.
m?RSU gt m?SUR
m?USV gt m?SUV
m?RSU m?USV gt m?SUR m?SUV
m?RSV gt m?RUV
Answer m?RSV gt m?RUV
16
Your Turn
Answer ?ABD gt ?DAB
Answer ?AED gt ?EAD
Answer ?EAB lt ?EDB
17
Theorem 5.10

• If one ? of a ? has a greater measure than
  another ?, then the side opposite the greater ?
  is longer than the side opposite the lesser ?.

A
AC gt BC gt CA
B
C
18
Example 4
19
Example 4
Theorem 5.10 states that if one side of a
triangle is longer than another side, then the
angle opposite the longer side has a greater
measure than the angle opposite the shorter side.
Since ?X is opposite the longest side it has the
greatest measure.
Answer So, Ebony should tie the ends marked Y
and Z.
20
Your Turn
KITE ASSEMBLY Tanya is following directions for
making a kite. She has two congruent triangular
pieces of fabric that need to be sewn together
along their longest side. The directions say to
begin sewing the two pieces of fabric together
at their smallest angles. At which two angles
should she begin sewing?
Answer ?A and ?D
21
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.

22
Example 1a
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 ?.
23
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.
24
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
25
Example 2
A 7 B 9 C 11 D 13
26
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.
27
Example 2
Graph the inequalities on the same number line.
28
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
29
Your Turn
A 4 B 9 C 12 D 16
Answer D