11 X1 T06 02 Triangle Theorems

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11 X1 T06 02 Triangle Theorems

  1. 1. Triangle Theorems
  2. 2. Triangle Theorems A The angle sum of any triangle is 180 B C
  3. 3. Triangle Theorems A The angle sum of any triangle is 180 B ∠A + ∠B + ∠C = 180 ( ∠sum ∆ABC = 180 ) C
  4. 4. Triangle Theorems A The angle sum of any triangle is 180 B ∠A + ∠B + ∠C = 180 ( ∠sum ∆ABC = 180 ) C Proof:
  5. 5. D Triangle Theorems A The angle sum of any triangle is 180 E B ∠A + ∠B + ∠C = 180 ( ∠sum ∆ABC = 180 ) C Proof: Construct DE||BC passing through A
  6. 6. D Triangle Theorems A The angle sum of any triangle is 180 E B ∠A + ∠B + ∠C = 180 ( ∠sum ∆ABC = 180 ) C Proof: Construct DE||BC passing through A ∠DAB = ∠ABC ( alternate ∠' s =, DE || BC )
  7. 7. D Triangle Theorems A The angle sum of any triangle is 180 E B ∠A + ∠B + ∠C = 180 ( ∠sum ∆ABC = 180 )  C Proof: Construct DE||BC passing through A ∠DAB = ∠ABC ( alternate ∠' s =, DE || BC ) ∠EAC = ∠ACB ( alternate ∠' s =, DE || BC )
  8. 8. D Triangle Theorems A The angle sum of any triangle is 180 E B ∠A + ∠B + ∠C = 180 ( ∠sum ∆ABC = 180 )  C Proof: Construct DE||BC passing through A ∠DAB = ∠ABC ( alternate ∠' s =, DE || BC ) ∠EAC = ∠ACB ( alternate ∠' s =, DE || BC ) ∠DAB + ∠BAC + ∠CAE = 180 ( straight ∠DAE = 180 ) 
  9. 9. D Triangle Theorems A The angle sum of any triangle is 180 E B ∠A + ∠B + ∠C = 180 ( ∠sum ∆ABC = 180 )  C Proof: Construct DE||BC passing through A ∠DAB = ∠ABC ( alternate ∠' s =, DE || BC ) ∠EAC = ∠ACB ( alternate ∠' s =, DE || BC ) ∠DAB + ∠BAC + ∠CAE = 180 ( straight ∠DAE = 180 )  ∴ ∠ABC + ∠BAC + ∠ACB = 180
  10. 10. A B C D
  11. 11. A The exterior angle of any triangle is equal to the sum of the two opposite B interior angles C D
  12. 12. A The exterior angle of any triangle is equal to the sum of the two opposite B interior angles ∠ACD = ∠A + ∠B ( exterior ∠, ∆CAB ) C D
  13. 13. A The exterior angle of any triangle is equal to the sum of the two opposite B interior angles ∠ACD = ∠A + ∠B ( exterior ∠, ∆CAB ) C D Proof:
  14. 14. A The exterior angle of any triangle is equal to the sum of the two opposite B E interior angles ∠ACD = ∠A + ∠B ( exterior ∠, ∆CAB ) C D Proof: Construct CE||BA
  15. 15. A The exterior angle of any triangle is equal to the sum of the two opposite B E interior angles ∠ACD = ∠A + ∠B ( exterior ∠, ∆CAB ) C D Proof: Construct CE||BA ∠ABC = ∠ECD ( corresponding ∠' s =, CE || BA)
  16. 16. A The exterior angle of any triangle is equal to the sum of the two opposite B E interior angles ∠ACD = ∠A + ∠B ( exterior ∠, ∆CAB ) C D Proof: Construct CE||BA ∠ABC = ∠ECD ( corresponding ∠' s =, CE || BA) ∠BAC = ∠ACE ( alternate ∠' s =, CE || BA)
  17. 17. A The exterior angle of any triangle is equal to the sum of the two opposite B E interior angles ∠ACD = ∠A + ∠B ( exterior ∠, ∆CAB ) C D Proof: Construct CE||BA ∠ABC = ∠ECD ( corresponding ∠' s =, CE || BA) ∠BAC = ∠ACE ( alternate ∠' s =, CE || BA) ∠ACD = ∠ACE + ∠ECD ( common ∠)
  18. 18. A The exterior angle of any triangle is equal to the sum of the two opposite B E interior angles ∠ACD = ∠A + ∠B ( exterior ∠, ∆CAB ) C D Proof: Construct CE||BA ∠ABC = ∠ECD ( corresponding ∠' s =, CE || BA) ∠BAC = ∠ACE ( alternate ∠' s =, CE || BA) ∠ACD = ∠ACE + ∠ECD ( common ∠) ∴ ∠ACD = ∠ABC + ∠BAC
  19. 19. A Polygon Theorems B C D
  20. 20. A Polygon Theorems B The angle sum of any quadrilateral is 360 C D
  21. 21. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 )  C D
  22. 22. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 )  Proof: C D
  23. 23. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 )  Proof: C D
  24. 24. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 )  Proof: ∠sum ∆ABC = 180 C D
  25. 25. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D
  26. 26. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360
  27. 27. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 B C A D E
  28. 28. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 The angle sum of any pentagon is 540 B C A D E
  29. 29. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 The angle sum of any pentagon is 540 B ∠A + ∠B + ∠C + ∠D + ∠E = 540 ( ∠sum ABCDE = 540 )  C A D E
  30. 30. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 The angle sum of any pentagon is 540 B ∠A + ∠B + ∠C + ∠D + ∠E = 540 ( ∠sum ABCDE = 540 )  C A Proof: D E
  31. 31. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 The angle sum of any pentagon is 540 B ∠A + ∠B + ∠C + ∠D + ∠E = 540 ( ∠sum ABCDE = 540 )  C A Proof: D E
  32. 32. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 The angle sum of any pentagon is 540 B ∠A + ∠B + ∠C + ∠D + ∠E = 540 ( ∠sum ABCDE = 540 )  C A Proof: ∠sum ∆ABE = 180 D E
  33. 33. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 The angle sum of any pentagon is 540 B ∠A + ∠B + ∠C + ∠D + ∠E = 540 ( ∠sum ABCDE = 540 )  C A Proof: ∠sum ∆ABE = 180 ∠sum ∆BED = 180 D E
  34. 34. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 The angle sum of any pentagon is 540 B ∠A + ∠B + ∠C + ∠D + ∠E = 540 ( ∠sum ABCDE = 540 ) C A Proof: ∠sum ∆ABE = 180 (+) ∠sum ∆BED = 180 D ∠sum ∆BDC = 180 E
  35. 35. A Polygon Theorems B The angle sum of any quadrilateral is 360 ∠A + ∠B + ∠C + ∠D = 360 ( ∠sum ABCD = 360 ) Proof: ∠sum ∆ABC = 180 (+) C ∠sum ∆ADC = 180 D ∠sum ABCD = 360 The angle sum of any pentagon is 540 B ∠A + ∠B + ∠C + ∠D + ∠E = 540 ( ∠sum ABCDE = 540 ) C A Proof: ∠sum ∆ABE = 180 (+) ∠sum ∆BED = 180 D ∠sum ∆BDC = 180 E ∠sum ABCDE = 540
  36. 36. The angle sum of any polygon is 180( n-2 ) ,  where n is the number of sides
  37. 37. The angle sum of any polygon is 180( n-2 ) ,  where n is the number of sides a e b c d
  38. 38. The angle sum of any polygon is 180( n-2 ) ,  where n is the number of sides a e b The exterior angle sum of any polygon is 360 c d
  39. 39. The angle sum of any polygon is 180( n-2 ) ,  where n is the number of sides a e b The exterior angle sum of any polygon is 360 c d a + b + c + d + e = 360 ( exterior∠ sum = 360 ) 
  40. 40. The angle sum of any polygon is 180( n-2 ) ,  where n is the number of sides a e b The exterior angle sum of any polygon is 360 c d a + b + c + d + e = 360 ( exterior∠ sum = 360 )  Exercise 8B; 1dg, 2c, 3dh, 5ace, 6ab (iii), 7b, 8bfh, 9ad, 10dh, 11ad, 12c, 16, 18, 20

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