Chapter 7

Geometric Inequalities

Chin-Sung Lin

Inequality Postulates

Mr. Chin-Sung Lin

Basic Inequality Postulates

Mr. Chin-Sung Lin

Comparison (Whole-Parts) Postulate

Transitive Property

Substitution Postulate

Trichotomy Postulate

Basic Inequality Postulates

Mr. Chin-Sung Lin

Addition Postulate

Subtraction Postulate

Multiplication Postulate

Division Postulate

Comparison Postulate

Mr. Chin-Sung Lin

A whole is greater than any of its parts

If a = b + c and a, b, c > 0

then a > b and a > c

Transitive Property

Mr. Chin-Sung Lin

If a, b, and c are real numbers such that a > b and b > c, then a > c

Substitution Postulate

Mr. Chin-Sung Lin

A quantity may be substituted for its equal in any statement of inequality

If a > b and b = c, then a > c

Trichotomy Postulate

Mr. Chin-Sung Lin

Give any two quantities, a and b, one and only one of the following is true:

a < b or a = b or a > b

Addition Postulate I

Mr. Chin-Sung Lin

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

Addition Postulate II

Mr. Chin-Sung Lin

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

Subtraction Postulate

Mr. Chin-Sung Lin

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

Multiplication Postulate I

Mr. Chin-Sung Lin

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

Multiplication Postulate II

Mr. Chin-Sung Lin

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

Division Postulate I

Mr. Chin-Sung Lin

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

Division Postulate II

Mr. Chin-Sung Lin

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

Theorems of Inequality

Mr. Chin-Sung Lin

Theorems of Inequality

Mr. Chin-Sung Lin

Exterior Angle Inequality Theorem

Greater Angle Theorem

Longer Side Theorem

Triangle Inequality Theorem

Converse of Pythagorean Theorem

Exterior Angle Inequality Theorem

Mr. Chin-Sung Lin

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 Prove: m1 > mA

m1 > mB

CA

B

1

Exterior Angle Inequality Theorem

Mr. Chin-Sung Lin

Statements Reasons

1. 1 is exterior angle and A & 1. Given B are remote interior angles

2. m1 = mA +mB 2. Exterior angle theorem

3. mA > 0 and mB > 0 3. Definition of triangles 4. m1 > mA 4. Comparison postulate m1 > mB

CA

B

1

Longer Side Theorem

Mr. Chin-Sung Lin

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)

Given: ∆ ABC with AC > BC Prove: mB > mA

B

C

A

B

C

A

D1

2

3

Longer Side Theorem

Mr. Chin-Sung Lin

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)

Given: ∆ ABC with AC > BC Prove: mB > mA

Longer Side Theorem

Mr. Chin-Sung Lin

Statements Reasons

1. AC > BC 1. Given2. Choose D on AC, CD = BC and 2. Form an isosceles triangle

draw a line segment BD

3. m1 = m2 3. Base angle theorem 4. m2 > mA 4. Exterior angle is greater

than the remote int. angle

5. m1 > mA 5. Substitution postulate6. mB = m1 + m3 6. Partition property7. mB > m1 7. Comparison postulate8. mB > mA 8. Transitive property

B

C

A

D1

2

3

Greater Angle Theorem

Mr. Chin-Sung Lin

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 mB > mAProve: AC > BC

B

C

A

Greater Angle Theorem

Mr. Chin-Sung Lin

Statements Reasons

1. mB > mA 1. Given2. Assume AC ≤ BC 2. Assume the opposite is

true3. mB = mA (when AC = BC) 3. Base angle theorem 4. mB < mA (when AC < BC) 4. Greater angle is opposite the

longer side 5. Statement 3 & 4 both contraidt 5. Contradicts to the given statement 16. AC > BC 6. The opposite of the

assumption is true

B

C

A

Triangle Inequality Theorem

Mr. Chin-Sung Lin

The sum of the lengths of any two sides of a triangle is greater than the length of the third side

Given: ∆ ABCProve: AB + BC > CA

B

C

A

Triangle Inequality Theorem

Mr. Chin-Sung Lin

The sum of the lengths of any two sides of a triangle is greater than the length of the third side

Given: ∆ ABCProve: AB + BC > CA

B

C

AD

1

Triangle Inequality Theorem

Mr. Chin-Sung Lin

Statements Reasons

1. Let D on AB and DB = CB, 1. Form an isosceles triangle and connect DC

2. m1 = mD 2. Base angle theorem 3. mDCA = m1 + mC 3. Partition property4. mDCA > m1 4. Comparison postulate 5. mDCA > mD 5. Substitution postulate6. AD > CA 6. Longer side is opposite the

greater angle 7. AD = AB + BD 7. Partition property8. AB + BD > CA 8. Substitution postulate9. AB + BC > CA 8. Substitution postulate

B

C

AD

1

Converse of Pythagorean Theorem

Mr. Chin-Sung Lin

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 sideProve: 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

B

C

A

Triangle Inequality Exercises

Mr. Chin-Sung Lin

Exercise 1

Mr. Chin-Sung Lin

∆ ABC with AB = 10, BC = 8, find the possible range of CA

Exercise 2

Mr. Chin-Sung Lin

List all the line segments from longest to shortest

C

D

A

B

60o

60o

61o

61o

59o

59o

Exercise 3

Mr. Chin-Sung Lin

Given the information in the diagram, if BD > BC, find the possible range of m3 and mB

C

DA B

30o 1 2

330o

Exercise 4

Mr. Chin-Sung Lin

∆ 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

Exercise 5

Mr. Chin-Sung Lin

∆ 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

Exercise 6

Mr. Chin-Sung Lin

Given: AC = ADProve: m2 > m1

A

C

BD

12

3

The End

Mr. Chin-Sung Lin