Objective: 5.3 & 5.5-5.6
Inequalities in One/Two Triangle(s)
Warm
Up:
_________&
The Triangle Inequality
Solve the inequality:
1. x + 3 < 14
2. 12 > 10 – x
3. 2x + 3 < 4x – 9
Find the measure of the third angle of
a triangle with the two given angle
measures.
4. 28°, 59°
5. x°, 2x°
Section 5.3
Inequalities in One Triangle
Section 5.5
The Triangle Inequality
Section 5.6
Inequalities in Two Triangles
Theorem 5.8 - Exterior Angle Inequality
• The measure of an exterior angle of a
triangle is greater than the measure of
either of the two nonadjacent interior
angles.
• m1 > mA and m1 > mB
A
1
C
B
Theorem 5.9
• If one side of a triangle is longer than another side, then
the angle opposite the longer side is larger than the angle
opposite the shorter side.
B
3
5
A
C
If BC > AB, then m A > m  C
Ex 1: Comparing Angles
• A landscape architect is designing a
triangular deck. She wants to place
benches in the two larger corners. Which
corners have the larger angles?
A
27ft
C
18ft
21ft
B
m C < m A < m B
Theorem
• If one angle of triangle is larger than another angle, then
the opposite side of the greater angle is longer than the
opposite side of the smaller angle.
F
60
D
40
E
If <D > <E, then, EF > DF
Ex 2: Comparing Sides
• Which side is the shortest?
T
U 52
62
V
Y
60
X
40
Z
Triangle Inequality Theorem
• The sum of the lengths of any two
sides of a triangle is greater than
the length of the third side.
AB + BC > AC
A
AC + BC > AB
AB + AC > BC
B
C
Ex 3: Can a triangle have sides
with the given lengths?
• 3ft, 2ft, 5ft
• 4cm, 2cm, 5cm
Ex 4: Finding Possible Side Lengths
• A triangle has side lengths of 10cm and 14cm.
Describe the possible lengths of the third side.
Let x represent the length of the third side.
Using the Triangle Inequality, write and solve inequalities
The length of the third side must be
greater than 4cm and less than 24cm.
List the angles in order of Triangle ABC from least to greatest if AB=2x+5,
AC=3x-10, BC=x+25 and the perimeter of Triangle ABC is 50.
Hinge Theorem/SAS Inequality
• If two sides of one triangle are
congruent to two sides of another
triangle, and the included angle of the
first is larger than the included angle of
the second, then the third side of the
first triangle is longer than the third
side of the second triangle.
RT > VX
Converse of the Hinge Theorem/SSS
Inequality
• If two sides of one triangle are congruent
to two sides of another triangle, and the
third side of the first is longer than the
third side of the second, then the included
angle of the first is larger than the
included angle of the second.
mA > mD
Ex 3: Finding Possible Side
Lengths and Angle Measures
Using the Hinge Theorem
and its converse, choose
possible side lengths or
angle measures from a
given list.
a. AB ≅ DE, BC ≅ EF,
AC = 12 inches, mB = 36°, and mE
= 80°. Which of the following is a
possible length for DF?
8 in., 10 in., 12 in., or 23 in.?
Ex 3 cont’d:
b. Given ∆RST and ∆XYZ, RT ≅ XZ, ST ≅ YZ, RS =
3.7 cm., XY = 4.5 cm, and mZ = 75°. Which
of the following is a possible measure for T:
60°, 75°, 90°, or 105°.