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

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 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/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: m1 > mA m1 > mB C A B 1

Exterior Angle Inequality Theorem Mr. Chin-Sung Lin StatementsReasons 1. 1 is exterior angle and A & 1. Given B are remote interior angles 2. m1 = mA +mB 2. Exterior angle theorem 3. mA > 0 and mB > 0 3. Definition of triangles 4. m1 > mA 4. Comparison postulate m1 > mB C A 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: mB > mA B C A

B C A D 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: mB > mA

Longer Side Theorem Mr. Chin-Sung Lin StatementsReasons 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 3. Base angle theorem 4. m2 > mA 4. Exterior angle is greater than the remote int. angle 5. m1 > mA 5. Substitution postulate 6. mB = m1 + m3 6. Partition property 7. mB > m1 7. Comparison postulate 8. mB > mA 8. Transitive property B C A D 1 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 mB > mA Prove: AC > BC B C A

Greater Angle Theorem Mr. Chin-Sung Lin StatementsReasons 1. mB > mA 1. Given 2. Assume AC ≤ BC 2. Assume the opposite is true 3. mB = mA (when AC = BC) 3. Base angle theorem 4. mB < mA (when AC < BC) 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 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: ∆ ABC Prove: 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: ∆ ABC Prove: AB + BC > CA B C A D 1

Triangle Inequality Theorem Mr. Chin-Sung Lin StatementsReasons 1. Let D on AB and DB = CB, 1. Form an isosceles triangle and connect DC 2. m1 = mD 2. Base angle theorem 3. mDCA = m1 + mC 3. Partition property 4. mDCA > m1 4. Comparison postulate 5. mDCA > mD 5. Substitution postulate 6. AD > CA 6. Longer side is opposite the greater angle 7. AD = AB + BD 7. Partition property 8. AB + BD > CA 8. Substitution postulate 9. AB + BC > CA 8. Substitution postulate B C A D 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 side Prove: If a 2 +b 2 = c 2, then the triangle is right If a 2 + b 2 > c 2, then the triangle is acute If a 2 + b 2 < c 2, 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 60 o 61 o 59 o

Exercise 3 Mr. Chin-Sung Lin Given the information in the diagram, if BD > BC, find the possible range of m3 and mB C D AB 30 o 12 3

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 = AD Prove: m2 > m1 A C B D 12 3

The End Mr. Chin-Sung Lin