# 5 5Inequalities of One Triangle

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```							             5.5 Inequalities
in One Triangle

Geometry
Mrs. Spitz
Fall, 2004
Objectives:
• Use triangle measurements to
decide which side is longest or which
angle is largest.
• Use the Triangle Inequality
Assignment
pp. 298-300 #1-25, 34
Objective 1: Comparing
Measurements of a Triangle
• In activity 5.5, you                                 largest angle
may have
discovered a
relationship
between the
positions of the
longest side
longest and
shortest sides of a
triangle and the
shortest side
position of its
angles.
The diagrams illustrate Thms. 5.10   smallest angle
and 5.11.
Theorem 5.10
If one side of a                  B

triangle is longer
than another side,       3
5
then the angle
opposite the
A                 C
longer side is
larger than the
angle opposite the
mA   > mC
shorter side.
Theorem 5.11
D
If one ANGLE of a                                   E
60°                40°
triangle is larger
than another
ANGLE, then the
SIDE opposite the
larger angle is                 F

longer than the
EF > DF
side opposite the
smaller angle.     You can write the measurements
of a triangle in order from least to
greatest.
Ex. 1: Writing Measurements in Order
from Least to Greatest

Write the
measurements
of the triangles               J

from least to                100°
greatest.
a. m G < mH <            45°
H
m J                                35°

JH < JG < GH                              G
Ex. 1: Writing Measurements in Order
from Least to Greatest

Write the
measurements               8
R

of the triangles
Q
from least to
greatest.
5       7

b. QP < PR < QR
m R < mQ < m                 P

P
Paragraph Proof – Theorem 5.10
A

Given►AC > AB
Prove ►mABC > mC
D
2

1
3
B                       C

Use the Ruler Postulate to locate a point D on AC
such that DA = BA. Then draw the segment BD.
In the isosceles triangle ∆ABD, 1 ≅ 2.
Because mABC = m1+m3, it follows that
mABC > m1. Substituting m2 for m1
produces mABC > m2. Because m2 = m3
+ mC, m2 > mC. Finally because mABC >
m2 and m2 > mC, you can conclude that
mABC > mC.
NOTE:
The proof of 5.10 in the slide previous
uses the fact that 2 is an exterior
angle for ∆BDC, so its measure is
the sum of the measures of the two
m2 must be greater than the
interior angle. This result is stated in
Theorem 5.12
Theorem 5.12-Exterior
Angle Inequality
• The measure of an exterior angle of
a triangle is greater than the
measure of either of the two non
• m1 > mA and m1 > mB
A

1
C             B
Ex. 2: Using Theorem 5.10

• DIRECTOR’S CHAIR. In the
director’s chair shown, AB ≅ AC and
BC > AB. What can you conclude
A

B                C
Ex. 2: Using Theorem 5.10
Solution
• Because AB ≅ AC,
∆ABC is isosceles, so       A

B ≅ C. Therefore,
mB = mC.
Because BC>AB,
mA > mC by
Theorem 5.10. By
substitution, mA >
C
can conclude that       B

mA >60°, mB< 60°,
and mC < 60°.
Objective 2: Using the
Triangle Inequality
• Not every group of three segments
can be used to form a triangle. The
lengths of the segments must fit a
certain relationship.
Ex. 3: Constructing a Triangle

a. 2 cm, 2 cm, 5 cm
b. 3 cm, 2 cm, 5 cm
c. 4 cm, 2 cm, 5 cm

Solution: Try drawing triangles with
the given side lengths. Only group
(c) is possible. The sum of the first
and second lengths must be
greater than the third length.
Ex. 3: Constructing a Triangle

a. 2 cm, 2 cm, 5 cm
b. 3 cm, 2 cm, 5 cm          2
2
c. 4 cm, 2 cm, 5 cm
5

C
D                               D
3                               4
2                               2

A       5               B       A           5           B
Theorem 5.13: Triangle Inequality

• The sum of the
lengths of any two
sides of a Triangle                   A

is greater than the
length of the third
side.
AB + BC > AC
C           B
AC + BC > AB
AB + AC > BC
Ex. 4: Finding Possible
Side Lengths
• A triangle has one      x + 10 > 14
side of 10 cm and       x>4
another of 14 cm.
Describe the possible
lengths of the third    10 + 14 > x
side                    24 > x
• SOLUTION: Let x
represent the length    ►So, the length of the
of the third side.        third side must be
Using the Triangle        greater than 4 cm and
Inequality, you can       less than 24 cm.
write and solve
inequalities.
#24 - homework
A                   • Solve the
inequality:
AB + AC > BC.
x+ 2                x+ 3

B          3x - 2      C   (x + 2) +(x + 3) > 3x – 2
2x + 5 > 3x – 2
5>x–2
7>x
5. Geography
AB + BC > AC
Masbate
MC + CG > MG
99 + 165 > x
264 > x        99 miles

x + 99 < 165                               Guiuan
165 miles

66 < x < 264

```
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