Download Inequalites in Triangles

The Triangle Inequality
&
Inequalities in Multiple
Triangles
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
Inequalities

An inequality simply shows a relationship between
any real numbers a and b such that if a > 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).
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.
Example 1:
Solve
Compare m3 to m1.
By the Exterior Angle Theorem,
m1 m3 m4. Since angle measures
are positive numbers and from the definition
of inequality, m1 > m3.
Compare m4 to m1.
By the Exterior Angle Theorem, m1 m3 m4.
By the definition of inequality, m1 > m4.
Compare m5 to m1.
Since all right angles are congruent, 4 5.
By the definition of congruent angles, m4 m5.
By substitution, m1 > m5.
Example 1:
Compare m2 to m5.
By the Exterior Angle Theorem, m5 m2 m3.
By the definition of inequality, m5 > m2.
Since we know that m1 > m5, by the
Transitive Property, m1 > m2.
Examine The results on the previous slides show that
m1 > m2, m1 > m3, m1 > m4, and
m1 > m5. Therefore, 1 has the greatest
measure.
Answer: 1 has the greatest measure.
Your Turn:
Determine which angle has the greatest measure.
Answer: 5 has the greatest measure.
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 > m 3
m 1 > m 4
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 > m4,
m14 > m11, m14 > m2, and m14 > m4 + m3.
Since 11 and 9 are vertical angles, they have equal
measure, so m14 > m9. m9 > m6 and m9 > m7,
so m14 > m6 and m14 > m7.
Answer: Thus, the measures of 4, 11, 9,  3,  2, 6,
and 7 are all less than m14 .
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 > m5,
and m16 > m10, so m16 > m5, m17 > m5 + m6,
m15 > m12, and m12 > m5, so m15 > m5.
Answer: Thus, the measures of 10, 16, 12, 15 and
17 are all greater than m5.
Your Turn:
Use the Exterior Angle Inequality Theorem to list all of
the angles that satisfy the stated condition.
a. all angles whose measures are less than m4
Answer: 5, 2, 8, 7
b. all angles whose measures are greater than m8
Answer: 4, 9, 5
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 .)
m 1 > m 2 > m 3
2
1
3
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 > mSUR.
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 < mSTV.
Example 3c:
Determine the relationship between the measures of
RSV and RUV.
mRSU > mSUR
mUSV > mSUV
mRSU + mUSV > mSUR + mSUV
mRSV > mRUV
Answer: mRSV > mRUV
Your Turn:
Determine the relationship between the measures of
the given angles.
a. ABD, DAB
Answer: ABD > DAB
b. AED, EAD
Answer: AED > EAD
c. EAB, EDB
Answer: EAB < EDB
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 > BC > CA
B
C
Example 4:
HAIR ACCESSORIES Ebony is following directions
for folding a handkerchief to make a bandana for her
hair. After she folds the handkerchief in half, the
directions tell her to tie the two smaller angles of the
triangle under her hair. If she folds the handkerchief
with the dimensions shown, which two ends should
she tie?
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.
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
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