Geometry Section 4-3

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Congruent Triangles

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Geometry Section 4-3

  1. 1. SECTION 4-3 Congruent Triangles
  2. 2. ESSENTIAL QUESTIONS How do you name and use corresponding parts of congruent polygons? How do you prove triangles congruent using the definition of congruence?
  3. 3. VOCABULARY 1. Congruent: 2. Congruent Polygons: 3. Corresponding Parts: Theorem 4.3 - Third Angles Theorem:
  4. 4. VOCABULARY 1. Congruent:Two figures with exactly the same size and shape 2. Congruent Polygons: 3. Corresponding Parts: Theorem 4.3 - Third Angles Theorem:
  5. 5. VOCABULARY 1. Congruent:Two figures with exactly the same size and shape 2. Congruent Polygons: All parts of one polygon are congruent to matching parts of another polygon 3. Corresponding Parts: Theorem 4.3 - Third Angles Theorem:
  6. 6. VOCABULARY 1. Congruent:Two figures with exactly the same size and shape 2. Congruent Polygons: All parts of one polygon are congruent to matching parts of another polygon 3. Corresponding Parts: The matching parts between two polygons; Corresponding parts have the same position in each polygon Theorem 4.3 - Third Angles Theorem:
  7. 7. VOCABULARY 1. Congruent:Two figures with exactly the same size and shape 2. Congruent Polygons: All parts of one polygon are congruent to matching parts of another polygon 3. Corresponding Parts: The matching parts between two polygons; Corresponding parts have the same position in each polygon Theorem 4.3 - Third Angles Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles must be congruent
  8. 8. CPCTC
  9. 9. CPCTC The Corresponding Parts of Congruent Triangles are Congruent
  10. 10. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.
  11. 11. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI
  12. 12. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH
  13. 13. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG
  14. 14. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG DE ≅ GF
  15. 15. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG DE ≅ GF EA ≅ FJ
  16. 16. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG DE ≅ GF EA ≅ FJ ∠A ≅ ∠J
  17. 17. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG DE ≅ GF EA ≅ FJ ∠A ≅ ∠J ∠B ≅ ∠I
  18. 18. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG DE ≅ GF EA ≅ FJ ∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H
  19. 19. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG DE ≅ GF EA ≅ FJ ∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H ∠D ≅ ∠G
  20. 20. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG DE ≅ GF EA ≅ FJ ∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H ∠D ≅ ∠G ∠E ≅ ∠F
  21. 21. EXAMPLE 1 Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. AB ≅ JI BC ≅ IH CD ≅ HG DE ≅ GF EA ≅ FJ ∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H ∠D ≅ ∠G ∠E ≅ ∠F Since all corresponding parts are congruent, ABCDE ≅ JIHGF
  22. 22. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
  23. 23. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. ∠P ≅ ∠O
  24. 24. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. 6y − 14 = 40 ∠P ≅ ∠O
  25. 25. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. 6y − 14 = 40 ∠P ≅ ∠O 6y = 54
  26. 26. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. 6y − 14 = 40 ∠P ≅ ∠O 6y = 54 y = 9
  27. 27. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. 6y − 14 = 40 ∠P ≅ ∠O 6y = 54 y = 9 IT ≅ NG
  28. 28. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. 6y − 14 = 40 ∠P ≅ ∠O 6y = 54 y = 9 x − 2 y = 7.5 IT ≅ NG
  29. 29. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. 6y − 14 = 40 ∠P ≅ ∠O 6y = 54 y = 9 x − 2 y = 7.5 IT ≅ NG x − 2(9) = 7.5
  30. 30. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. 6y − 14 = 40 ∠P ≅ ∠O 6y = 54 y = 9 x − 2 y = 7.5 IT ≅ NG x − 2(9) = 7.5 x − 18 = 7.5
  31. 31. EXAMPLE 2 In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y. 6y − 14 = 40 ∠P ≅ ∠O 6y = 54 y = 9 x − 2 y = 7.5 IT ≅ NG x − 2(9) = 7.5 x − 18 = 7.5 x = 25.5
  32. 32. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON
  33. 33. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON 1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
  34. 34. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON 1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given
  35. 35. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON 1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given 2. ∠LNM ≅ ∠PNO
  36. 36. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON 1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given 2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm.
  37. 37. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON 1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given 2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm. 3. ∠M ≅ ∠O
  38. 38. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON 1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given 2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm. 3. ∠M ≅ ∠O 3. Third Angle Theorem
  39. 39. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON 1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given 2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm. 3. ∠M ≅ ∠O 3. Third Angle Theorem 4.△LMN ≅△PON
  40. 40. EXAMPLE 3 Write a two-column proof. Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON Prove: △LMN ≅△PON 1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given 2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm. 3. ∠M ≅ ∠O 3. Third Angle Theorem 4.△LMN ≅△PON 4. Def. of congruent polygons
  41. 41. PROBLEM SET
  42. 42. PROBLEM SET p. 257 #1-23 odd, 29, 36, 39, 49, 53, 55 “I've always tried to go a step past wherever people expected me to end up.” - Beverly Sills

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