Geometry Section 4-2

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Geometry Section 4-2

  1. 1. SECTION 4-2 Angles ofTriangles
  2. 2. ESSENTIAL QUESTIONS • How do you apply theTriangle Angle-SumTheorem? • How do you apply the Exterior AngleTheorem?
  3. 3. VOCABULARY 1.Auxiliary Line: 2. Exterior Angle: 3. Remote Interior Angles: 4. Flow Proof:
  4. 4. VOCABULARY 1.Auxiliary Line: An extra line or segment that is added to a figure to help analyze geometric relationships 2. Exterior Angle: 3. Remote Interior Angles: 4. Flow Proof:
  5. 5. VOCABULARY 1.Auxiliary Line: An extra line or segment that is added to a figure to help analyze geometric relationships 2. Exterior Angle: Formed outside a triangle when one side of the triangle is extended;The exterior angle is adjacent to the interior angle of the triangle 3. Remote Interior Angles: 4. Flow Proof:
  6. 6. VOCABULARY 1.Auxiliary Line: An extra line or segment that is added to a figure to help analyze geometric relationships 2. Exterior Angle: Formed outside a triangle when one side of the triangle is extended;The exterior angle is adjacent to the interior angle of the triangle 3. Remote Interior Angles: The two interior angles that are not adjacent to a given exterior angle 4. Flow Proof:
  7. 7. VOCABULARY 1.Auxiliary Line: An extra line or segment that is added to a figure to help analyze geometric relationships 2. Exterior Angle: Formed outside a triangle when one side of the triangle is extended;The exterior angle is adjacent to the interior angle of the triangle 3. Remote Interior Angles: The two interior angles that are not adjacent to a given exterior angle 4. Flow Proof: Uses statements written in boxes with arrows to show a logical progression of an argument
  8. 8. THEOREMS & COROLLARIES 4.1 -Triangle Angle-SumTheorem: 4.2 - Exterior AngleTheorem: 4.1 Corollary: 4.2 Corollary:
  9. 9. THEOREMS & COROLLARIES 4.1 -Triangle Angle-SumTheorem: The sum of the measures of the angles of any triangle is 180° 4.2 - Exterior AngleTheorem: 4.1 Corollary: 4.2 Corollary:
  10. 10. THEOREMS & COROLLARIES 4.1 -Triangle Angle-SumTheorem: The sum of the measures of the angles of any triangle is 180° 4.2 - Exterior AngleTheorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles 4.1 Corollary: 4.2 Corollary:
  11. 11. THEOREMS & COROLLARIES 4.1 -Triangle Angle-SumTheorem: The sum of the measures of the angles of any triangle is 180° 4.2 - Exterior AngleTheorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles 4.1 Corollary: The acute angles of a right triangle are complementary 4.2 Corollary:
  12. 12. THEOREMS & COROLLARIES 4.1 -Triangle Angle-SumTheorem: The sum of the measures of the angles of any triangle is 180° 4.2 - Exterior AngleTheorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles 4.1 Corollary: The acute angles of a right triangle are complementary 4.2 Corollary: There can be at most one right or obtuse angle in a triangle
  13. 13. EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle.
  14. 14. EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43− 74
  15. 15. EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43− 74 = 63°
  16. 16. EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43− 74 = 63° m∠2
  17. 17. EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43− 74 = 63° m∠2 = 63°
  18. 18. EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43− 74 = 63° m∠2 = 63° m∠3 =180 − 63− 79
  19. 19. EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43− 74 = 63° m∠2 = 63° m∠3 =180 − 63− 79 = 38°
  20. 20. EXAMPLE 2 Find m∠FLW.
  21. 21. EXAMPLE 2 Find m∠FLW. m∠FLW = m∠LOW + m∠OWL
  22. 22. EXAMPLE 2 Find m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32
  23. 23. EXAMPLE 2 Find m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32
  24. 24. EXAMPLE 2 Find m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32 x = 80
  25. 25. EXAMPLE 2 Find m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32 x = 80 m∠FLW = 2(80)− 48
  26. 26. EXAMPLE 2 Find m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32 x = 80 m∠FLW = 2(80)− 48 =160 − 48
  27. 27. EXAMPLE 2 Find m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32 x = 80 m∠FLW = 2(80)− 48 =160 − 48 =112°
  28. 28. EXAMPLE 3 Find the measure of each numbered angle.
  29. 29. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41
  30. 30. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49°
  31. 31. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48
  32. 32. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42°
  33. 33. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42
  34. 34. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48°
  35. 35. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34
  36. 36. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 m∠2 =180 − 56 − 48
  37. 37. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 m∠2 =180 − 56 − 48
  38. 38. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 m∠2 =180 − 56 − 48 = 76°
  39. 39. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 m∠2 =180 − 56 − 48 = 76° m∠1=180 − 76
  40. 40. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 m∠2 =180 − 56 − 48 = 76° m∠1=180 − 76 =104°
  41. 41. EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 56° m∠2 =180 − 56 − 48 = 76° m∠1=180 − 76 =104°
  42. 42. PROBLEM SET
  43. 43. PROBLEM SET p. 248 #1-37 odd, 46, 57 “We rarely think people have good sense unless they agree with us.” - Francois de La Rochefoucauld

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