Geometry Section 5-3 11-12

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Geometry Section 5-3 11-12

  1. 1. SECTION 5-3 Inequalities in One TriangleTuesday, March 6, 2012
  2. 2. ESSENTIAL QUESTIONS How do you recognize and apply properties of inequalities to the measures of the angles of a triangle? How do you recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle?Tuesday, March 6, 2012
  3. 3. VOCABULARY 1. Inequality:Tuesday, March 6, 2012
  4. 4. VOCABULARY 1. Inequality: For any real numbers a and b, a > b IFF there is a positive number c such that a = b + cTuesday, March 6, 2012
  5. 5. PROPERTIES OF INEQUALITY FOR REAL NUMBERS 1. Comparison Property of Inequality: 2. Transitive Property of Inequality: 3. Addition Property of Inequality: 4. Subtraction Property of Inequality:Tuesday, March 6, 2012
  6. 6. PROPERTIES OF INEQUALITY FOR REAL NUMBERS 1. Comparison Property of Inequality: a < b, a = b, a > b 2. Transitive Property of Inequality: 3. Addition Property of Inequality: 4. Subtraction Property of Inequality:Tuesday, March 6, 2012
  7. 7. PROPERTIES OF INEQUALITY FOR REAL NUMBERS 1. Comparison Property of Inequality: a < b, a = b, a > b 2. Transitive Property of Inequality: If a < b and b < c, then a < c; If a > b and b > c, then a > c 3. Addition Property of Inequality: 4. Subtraction Property of Inequality:Tuesday, March 6, 2012
  8. 8. PROPERTIES OF INEQUALITY FOR REAL NUMBERS 1. Comparison Property of Inequality: a < b, a = b, a > b 2. Transitive Property of Inequality: If a < b and b < c, then a < c; If a > b and b > c, then a > c 3. Addition Property of Inequality: If a < b, then a + c < b + c; If a > b, then a + c > b + c 4. Subtraction Property of Inequality:Tuesday, March 6, 2012
  9. 9. PROPERTIES OF INEQUALITY FOR REAL NUMBERS 1. Comparison Property of Inequality: a < b, a = b, a > b 2. Transitive Property of Inequality: If a < b and b < c, then a < c; If a > b and b > c, then a > c 3. Addition Property of Inequality: If a < b, then a + c < b + c; If a > b, then a + c > b + c 4. Subtraction Property of Inequality: If a < b, then a − c < b − c; If a > b, then a − c > b − cTuesday, March 6, 2012
  10. 10. THEOREMS 5.8 - Exterior Angle Inequality: The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior anglesTuesday, March 6, 2012
  11. 11. ANGLE-SIDE RELATIONSHIPS IN TRIANGLES 5.9: If one side of a triangle is longer than another side, then the angle oppostie the longer side has a greater measure than the angle opposite the shorter side 5.10: If one angle of a triangle has a greater measure than another angle, then the side oppostie the larger angle has a greater measure than the side opposite the smaller angleTuesday, March 6, 2012
  12. 12. EXAMPLE 1 Use the Exterior Angles Inequality to list all of the angles that satisfy the stated condition. a. Measures less than m∠14 b. Measures greater than m∠5Tuesday, March 6, 2012
  13. 13. EXAMPLE 1 Use the Exterior Angles Inequality to list all of the angles that satisfy the stated condition. a. Measures less than m∠14 ∠7, ∠12, ∠1, ∠8, ∠10, ∠4, ∠5 b. Measures greater than m∠5Tuesday, March 6, 2012
  14. 14. EXAMPLE 1 Use the Exterior Angles Inequality to list all of the angles that satisfy the stated condition. a. Measures less than m∠14 ∠7, ∠12, ∠1, ∠8, ∠10, ∠4, ∠5 b. Measures greater than m∠5 ∠10, ∠16, ∠12, ∠14, ∠15Tuesday, March 6, 2012
  15. 15. EXAMPLE 2 List the angles of ∆ABC in order from smallest to largest.Tuesday, March 6, 2012
  16. 16. EXAMPLE 2 List the angles of ∆ABC in order from smallest to largest. ∠C, ∠A, ∠BTuesday, March 6, 2012
  17. 17. EXAMPLE 3 List the sides of ∆ABC in order from shortest to longest.Tuesday, March 6, 2012
  18. 18. EXAMPLE 3 List the sides of ∆ABC in order from shortest to longest. AC, AB, BCTuesday, March 6, 2012
  19. 19. EXAMPLE 4 Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?Tuesday, March 6, 2012
  20. 20. EXAMPLE 4 Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie? ∠Y and ∠Z should be tied, as they are the smallest angles (opposite shortest sides)Tuesday, March 6, 2012
  21. 21. CHECK YOUR UNDERSTANDING Review problems #1-7 on p. 346Tuesday, March 6, 2012
  22. 22. PROBLEM SETTuesday, March 6, 2012
  23. 23. PROBLEM SET p. 346 #9-37 odd, 45, 53, 57 “Health is not simply the absence of sickness.” - Hannah GreenTuesday, March 6, 2012

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