Chapter 7 Geometric Inequalities

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## Chapter 7 Geometric Inequalities

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**Chapter 7Geometric 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 SubtractionPostulate MultiplicationPostulate DivisionPostulate 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 < band c < d, then a + c < b +d Mr. Chin-Sung Lin**SubtractionPostulate**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**B**1 A C 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 Exterior Angle Inequality Theorem Mr. Chin-Sung Lin**B**1 A C Exterior Angle Inequality Theorem 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. Comparisonpostulate m1 > mB Mr. Chin-Sung Lin**C**B A 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**C**D 2 1 3 B A 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**C**D 2 1 3 B A Statements 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. m1 = m2 3. Base angle theorem 4. m2 > mA 4. Exterior angle is greater than the remote int. angle 5. m1 > mA 5. Substitutionpostulate 6. mB = m1 + m3 6. Partition property 7. mB > m1 7. Comparisonpostulate 8. mB > mA 8. Transitive property Longer Side Theorem Mr. Chin-Sung Lin**C**B A 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 > mA Prove: AC > BC Greater Angle Theorem Mr. Chin-Sung Lin**C**B A Statements Reasons 1. mB > mA 1. Given 2. Assume AC ≤ BC 2. Assume the opposite is true 3. 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 1 6. AC > BC 6. The opposite of the assumption is true Greater Angle Theorem Mr. Chin-Sung Lin**C**A B 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 Triangle Inequality Theorem Mr. Chin-Sung Lin**C**A B 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 Triangle Inequality Theorem 1 D Mr. Chin-Sung Lin**C**A D B 1 Triangle Inequality Theorem 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 property 4. mDCA > m1 4. Comparisonpostulate 5. mDCA > mD 5. Substitutionpostulate 6. AD > CA 6. Longer side is opposite the greater angle 7. AD = AB + BD 7. Partition property 8. AB + BD > CA 8. Substitutionpostulate 9. AB + BC > CA 8. Substitutionpostulate Mr. Chin-Sung Lin**C**B A 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 Converse of Pythagorean Theorem Mr. Chin-Sung Lin**Triangle Inequality Exercises**Mr. Chin-Sung Lin**∆ ABC with AB = 10, BC = 8, find the possible range of CA**Exercise 1 Mr. Chin-Sung Lin**D**59o 60o 61o A C 60o 59o 61o B List all the line segments from longest to shortest Exercise 2 Mr. Chin-Sung Lin**C**3 30o 30o 1 2 A B D Given the information in the diagram, if BD > BC, find the possible range of m3 and mB Exercise 3 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 4 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 5 Mr. Chin-Sung Lin**C**2 1 3 A B D Given: AC = AD Prove: m2 > m1 Exercise 6 Mr. Chin-Sung Lin**The End**Mr. Chin-Sung Lin