Download Chapter 5: Relationships Within Triangles

Chapter 5: Relationships
Within Triangles
5.5
Inequalities in Triangles
Comparison Property of Equality
• If a = b + c, and c > 0, then a > b.
b
a
c
Corollary
• Corollary to the Triangle Exterior Angle
Theorem:
– The measure of an exterior angle of a triangle is
greater than the measure of each of its remote
interior angles.
m1  m2
m1  m3
2
3
1
Theorem 5-10
• If two sides of a triangle are not congruent, then
the larger angle lies opposite the longer side.
B
AC  BC  AB
so
B  A  C
A
C
Example 2
• 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
27 ft
21 ft
C
18 ft
B
Theorem 5-11
If two angles of a triangle are not congruent, then
the larger side lies opposite the larger angle.
B
B  A  C
so
AC  BC  AB
A
C
Example 3
• List the sides of ΔTUV in order from shortest to
longest:
T
58
U
62
V
Example 3a
• List the sides of ΔXYZ in order from shortest to
longest. Explain your reasoning!
Y
40
X
60
Z
Theorem 5-12
• Triangle Inequality Theorem
– The sum of the lengths of any two sides of a
triangle is greater than the length of the third side.
Y
XZ  XY  ZY
XY  YZ  XZ
XZ  YZ  XY
X
Z
Example 4
• Can a triangle have sides with the given lengths?
• 3 ft, 7 ft, 8 ft
• 3 cm, 6 cm, 10 cm
Example 5
• A triangle has sides of lengths 8 cm and 10 cm.
Describe the lengths possible for the third side.
Example 5a
• A triangle has sides of lengths 3 in and 12 in.
Describe the lengths possible for the third side.
Homework