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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
mA > mC
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 < mH < 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 < mQ < m P
P
Paragraph Proof – Theorem 5.10
A
Given►AC > AB
Prove ►mABC > mC
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 mABC = m1+m3, it follows that
mABC > m1. Substituting m2 for m1
produces mABC > m2. Because m2 = m3
+ mC, m2 > mC. Finally because mABC >
m2 and m2 > mC, you can conclude that
mABC > mC.
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
nonadjacent interior angles. Then
m2 must be greater than the
measure of either nonadjacent
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
adjacent interior angles.
• m1 > mA and m1 > mB
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
about the angles in ∆ABC?
A
B C
Ex. 2: Using Theorem 5.10
Solution
• Because AB ≅ AC,
∆ABC is isosceles, so A
B ≅ C. Therefore,
mB = mC.
Because BC>AB,
mA > mC by
Theorem 5.10. By
substitution, mA >
mB. In addition, you
C
can conclude that B
mA >60°, mB< 60°,
and mC < 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
x < 66 Cadiz
66 < x < 264
