Chapter 7
Geometric Inequalities
Chin-Sung Lin
Inequality Postulates
Mr. Chin-Sung Lin
Basic Inequality Postulates
Comparison (Whole-Parts) Postulate
Transitive Property
Substitution Postulate
Trichotomy Postulate
Mr. Chin-Sung Lin
Basic Inequality Postulates
Addition Postulate
Subtraction Postulate
Multiplication Postulate
Division Postulate
Mr. Chin-Sung Lin
Comparison Postulate
A whole is greater than any of its parts
If a = b + c and a, b, c > 0
then a > b and a > c
Mr. Chin-Sung Lin
Transitive Property
If a, b, and c are real numbers such that a > b and
b > c, then a > c
Mr. Chin-Sung Lin
Substitution Postulate
A quantity may be substituted for its equal in
any statement of inequality
If a > b and b = c, then a > c
Mr. Chin-Sung Lin
Trichotomy Postulate
Give any two quantities, a and b, one and only one
of the following is true:
a < b or a = b or a > b
Mr. Chin-Sung Lin
Addition Postulate I
If equal quantities are added to unequal quantities,
then the sum are unequal in the same order
If a > b, then a + c > b + c
If a < b, then a + c < b + c
Mr. Chin-Sung Lin
Addition Postulate II
If unequal quantities are added to unequal
quantities in the same order, then the sum are
unequal in the same order
If a > b and c > d, then a + c > b + d
If a < b and c < d, then a + c < b + d
Mr. Chin-Sung Lin
Subtraction Postulate
If equal quantities are subtracted from unequal
quantities, then the difference are unequal in
the same order
If a > b, then a - c > b - c
If a < b, then a - c < b - c
Mr. Chin-Sung Lin
Multiplication Postulate I
If unequal quantities are multiplied by positive equal
quantities, then the products are unequal in the
same order
c > 0:
If a > b, then ac > bc
If a < b, then ac < bc
Mr. Chin-Sung Lin
Multiplication Postulate II
If unequal quantities are multiplied by negative
equal quantities, then the products are unequal
in the opposite order
c < 0:
If a > b, then ac < bc
If a < b, then ac > bc
Mr. Chin-Sung Lin
Division Postulate I
If unequal quantities are divided by positive equal
quantities, then the quotients are unequal in the
same order
c > 0:
If a > b, then a/c > b/c
If a < b, then a/c < b/c
Mr. Chin-Sung Lin
Division Postulate II
If unequal quantities are divided by negative equal
quantities, then the quotients are unequal in the
opposite order
c < 0:
If a > b, then a/c < b/c
If a < b, then a/c > b/c
Mr. Chin-Sung Lin
Theorems of Inequality
Mr. Chin-Sung Lin
Theorems of Inequality
Exterior Angle Inequality Theorem
Greater Angle Theorem
Longer Side Theorem
Triangle Inequality Theorem
Converse of Pythagorean Theorem
Mr. Chin-Sung Lin
Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle
is always greater than the measure of either
remote interior angle
Given: ∆ ABC with exterior angle 1
B
Prove: m1 > mA
m1 > mB
1
A
C
Mr. Chin-Sung Lin
Exterior Angle Inequality Theorem
B
A
Statements
1
C
Reasons
1. 1 is exterior angle and A &
B are remote interior angles
1. Given
2. m1 = mA +mB
2. Exterior angle theorem
3. mA > 0 and mB > 0
4. m1 > mA
m1 > mB
3. Definition of triangles
4. Comparison postulate
Mr. Chin-Sung Lin
Longer Side Theorem
If the length of one side of a triangle is longer
than the length of another side, then the
measure of the angle opposite the longer
side is greater than that of the angle
opposite the shorter side (In a triangle the
greater angle is opposite the longer side)
C
Given: ∆ ABC with AC > BC
Prove: mB > mA
B
A
Mr. Chin-Sung Lin
Longer Side Theorem
If the length of one side of a triangle is longer
than the length of another side, then the
measure of the angle opposite the longer
side is greater than that of the angle
opposite the shorter side (In a triangle the
greater angle is opposite the longer side)
C
Given: ∆ ABC with AC > BC
Prove: mB > mA
2
1
B
3
D
A
Mr. Chin-Sung Lin
C
Longer Side Theorem
Statements
1
B
2
3
D
A
Reasons
1. AC > BC
1. Given
2. Choose D on AC, CD = BC and 2. Form an isosceles triangle
draw a line segment BD
3. m1 = m2
4. m2 > mA
5.
6.
7.
8.
m1
mB
mB
mB
> mA
= m1 + m3
> m1
> mA
3. Base angle theorem
4. Exterior angle is greater
than the remote int. angle
5. Substitution postulate
6. Partition property
7. Comparison postulate
8. Transitive property
Mr. Chin-Sung Lin
Greater Angle Theorem
If the measure of one angle of a triangle is
greater than the measure of another angle,
then the side opposite the greater angle is
longer than the side opposite the smaller
angle (In a triangle the longer side is
opposite the greater angle)
Given: ∆ ABC with mB > mA
Prove: AC > BC
B
C
A
Mr. Chin-Sung Lin
C
Greater Angle Theorem B
Statements
1.
2.
3.
4.
A
Reasons
mB > mA
Assume AC ≤ BC
mB = mA (when AC = BC)
mB < mA (when AC < BC)
1. Given
2. Assume the opposite is true
3. Base angle theorem
4. Greater angle is opposite the
longer side
5. Statement 3 & 4 both contraidt 5. Contradicts to the given
statement 1
6. AC > BC
6. The opposite of the
assumption is true
Mr. Chin-Sung Lin
Triangle Inequality Theorem
The sum of the lengths of any two sides of a
triangle is greater than the length of the
third side
Given: ∆ ABC
Prove: AB + BC > CA
C
B
A
Mr. Chin-Sung Lin
Triangle Inequality Theorem
The sum of the lengths of any two sides of a
triangle is greater than the length of the
third side
Given: ∆ ABC
Prove: AB + BC > CA
C
1
D
B
A
Mr. Chin-Sung Lin
Triangle Inequality Theorem
C
1
Statements
D
1. Let D on AB and DB = CB,
and connect DC
2. m1 = mD
3. mDCA = m1 + mC
4. mDCA > m1
5. mDCA > mD
6. AD > CA
7. AD = AB + BD
8. AB + BD > CA
9. AB + BC > CA
A
B
Reasons
1. Form an isosceles triangle
2.
3.
4.
5.
6.
Base angle theorem
Partition property
Comparison postulate
Substitution postulate
Longer side is opposite the
greater angle
7. Partition property
8. Substitution postulate
8. Substitution postulate
Mr. Chin-Sung Lin
Converse of Pythagorean Theorem
A corollary of the Pythagorean theorem's converse is
a simple means of determining whether a triangle
is right, obtuse, or acute
Given: ∆ ABC and c is the longest side
Prove: If a2 +b2 = c2, then the triangle is right
If a2 + b2 > c2, then the triangle is acute
If a2 + b2 < c2, then the triangle is obtuse
C
B
A
Mr. Chin-Sung Lin
Triangle Inequality
Exercises
Mr. Chin-Sung Lin
Exercise 1
∆ ABC with AB = 10, BC = 8, find the possible
range of CA
Mr. Chin-Sung Lin
Exercise 2
List all the line segments from longest to
shortest
D
59o
A
60o
61o
59o
60o
C
61o
B
Mr. Chin-Sung Lin
Exercise 3
Given the information in the diagram,
if BD > BC, find the possible range
of m3 and mB
C
30o
A
30o
1
2
D
3
B
Mr. Chin-Sung Lin
Exercise 4
∆ ABC with AB = 5, BC = 3, CA = 7,
(a) what’s the type of ∆ ABC ? (Obtuse ∆?
Acute ∆? Right ∆?)
(b) list the angles of the triangle from
largest to smallest
Mr. Chin-Sung Lin
Exercise 5
∆ ABC with AB = 5, BC = 3,
(a) if ∆ ABC is a right triangle, find the
possible values of CA
(b) if ∆ ABC is a obtuse triangle, find the
possible range of CA
(c) if ∆ ABC is a acute triangle, find the
possible range of CA
Mr. Chin-Sung Lin
Exercise 6
Given: AC = AD
Prove: m2 > m1
C
2
A
1
3
D
B
Mr. Chin-Sung Lin
The End
Mr. Chin-Sung Lin