Chapter 7 Geometric Inequalities Chin-Sung Lin
Jan 18, 2016
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