Download 2. If c = d, then a + c > b + d (Algebra)

Geometry 5.2 INEQUALITIES Sides/Angles of Triangles
Objectives:

Learn the Exterior Angle INEQUALITY Theorem
 To Recognize the LONGEST Side from the
Largest Angle
 To Recognize the LARGEST Angle from the
Longest Side
A REMINDER about INEQUALITIES
If a > b if there exists a positive number c such that
b+c=a
If a < b if there exists a positive number c such that
a+c=b
EXTERIOR ANGLE INEQUALITY Theorem
The measure of an EXTERIOR Angle of a Triangle
is GREATER Than
the measure of EITHER of its REMOTE INTERIOR Angles.
EXTERIOR ANGLE INEQUALITY Theorem
The measure of an EXTERIOR Angle of a Triangle
is GREATER Than
the measure of EITHER of its REMOTE INTERIOR Angles.
C
3
2
A
1
B
D
INEQUALITY Algebra
Assume: a > b
1.
If b > c, then a > c
(Transitive)
INEQUALITY Algebra
Assume: a > b
1.
If b > c, then a > c
(Transitive)
2.
If c = d, then a + c > b + d
(Algebra)
INEQUALITY Algebra
Assume: a > b
1.
If b > c, then a > c
(Transitive)
2.
If c = d, then a + c > b + d
3.
If c > d, then a + c > b + d (Algebra)
(Algebra)
INEQUALITY Algebra
Assume: a > b
1.
If b > c, then a > c
(Transitive)
2.
If c = d, then a + c > b + d
3.
If c > d, then a + c > b + d (Algebra)
4.
If b = c, then a > c
(Algebra)
(Substitution)
LONGEST SIDE - BIGGEST ANGLE Theorem
IF
ONE SIDE of a Triangle is LONGER than another side,
THEN
the Angle OPPOSITE the Longer Side is GREATER
than the Angle OPPOSITE the Shorter Side.
LONGEST SIDE - BIGGEST ANGLE Th.
R
P
Q
Given:  RPQ with RQ  RP
Prove: m  RPQ  m  Q
WITHOUT using the BIGGEST Angle LONGEST Side Theorem
TRICHOTOMY Property
For any TWO Real Numbers a and b,
Only ONE of THREE Possible Relationships Exists:
a < b
or
a = b
or
a > b
BIGGEST ANGLE - LONGEST SIDE Th.
IF
ONE Angle of a Triangle is GREATER THAN a
SECOND Angle,
THEN
The Side OPPOSITE the GREATER Angle is LONGER
than the Side OPPOSITE the Smaller Angle.
BIGGEST ANGLE - LONGEST SIDE Th.
R
P
Given:  RPQ with m  P  m  Q
Prove: RQ
>
RP
Q
HUGE
D
58
A
56
63
59
70
B
Name the LONGEST Segment.
54
C
S
60 65
R
55
P
70
50 60
Q
The LONGEST Segment is?
R
1
P
Given: m  Q   P
RP  R S
S
Prove:
Q
mQ  m1
D
C
1
3
2
B
A
Given: DC || AB
m 3  m1
Prove: BD > AD
Geometry 5.2
You should be able to:
 State the Exterior Angle INEQUALITY Theorem
 Identify the LONGEST Side from the
Largest Angle
 Identify the LARGEST Angle from the
LONGEST Side