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Chapter 4 Congruent Triangles

Chapter 4 Congruent Triangles

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Chapter 4 Congruent Triangles

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  1. Chapter 4Congruent Triangles • Identify the corresponding parts of congruent figures • Prove two triangles are congruent • Apply the theorems and corollaries about isosceles triangles

  2. 4.1 Congruent Figures Objectives • Identify the corresponding parts of congruent figures

  3. What we already know… • Congruent Segments • Same length • Congruent Angles • Same degree measure

  4. F E Congruent Figures Exactly the same size and shape. Don’t ASSume ! C B A D

  5. Two figures are congruent if corresponding vertices can be matched up so that: 1. All corresponding sides are congruent 2. All corresponding angles are congruent. Definition of Congruency

  6. What does corresponding mean again? • Matching • In the same position

  7. Volunteer • Draw a large scalene triangle (with a ruler) • Cut out two congruent triangles that are the same • Label the Vertices A, B, C and D, E, F

  8. You can slide and rotate the triangles around so that they MATCH up perfectly. ABC DEF A E C B F D

  9. The order in which you name the triangles matters ! ABC DEF A E C B F D

  10. Based on the definition of congruency…. • Three pairs of corresponding sides • Three pairs of corresponding angles 1.  A   D 1. AB  DE 2.  B   E 2. BC  EF 3.  C   F 3. CA  FD

  11. It is not practical to cut out and move the triangles around

  12.  ABC   XYZ • Means that the letters X and A, which appear first, name corresponding vertices and that •  X   A. • The letters Y and B come next, so •  Y   B and • XY  AB

  13. CAUTION !! Don't ASSume • If the diagram doesn’t show the markings or • You don’t have a reason • Shared sides, shared angles, vertical angles, parallel lines

  14. White Boards • Suppose  TIM   BER IM  ___

  15. White Boards • Suppose  TIM   BER IM  ER , Why ?

  16. White Boards • Corresponding Parts of Congruent Triangles are Congruent

  17. White Boards • Suppose  TIM   BER ___   R

  18. White Boards • Suppose  TIM   BER  M   R, Why?

  19. White Boards • Corresponding Parts of Congruent Triangles are Congruent

  20. White Boards • Suppose  TIM   BER  MTI   ____

  21. White Boards • Suppose  TIM   BER  MTI   RBE

  22. White Boards • If  ABC   XYZ m  B = 80 m  C = 50 Name four congruent angles

  23. White Boards • If  ABC   XYZ m  B = 80 m  C = 50  A,  C ,  X,  Z

  24. White Boards • If  ABC   XYZ Write six congruences that must be correct

  25. White Boards • If  ABC   XYZ 1.  A   X 1. AB  XY 2.  B   Y 2. BC  YZ 3.  C   Z 3. CA  ZX

  26. Remote time • Always • Sometimes • Never • I don’t know

  27. A. AlwaysB. SometimesC. NeverD. I don’t know • An acute triangle is __________ congruent to an obtuse triangle.

  28. A. AlwaysB. SometimesC. NeverD. I don’t know • A polygon is __________ congruent to itself.

  29. A. AlwaysB. SometimesC. NeverD. I don’t know • A right triangle is ___________ congruent to another right triangle.

  30. A. AlwaysB. SometimesC. NeverD. I don’t know • If  ABC   XYZ,  A is ____________ congruent to  Y.

  31. A. AlwaysB. SometimesC. NeverD. I don’t know • If  ABC   XYZ,  B is ____________ congruent to  Y.

  32. A. AlwaysB. SometimesC. NeverD. I don’t know • If  ABC   XYZ, AB is ____________ congruent to ZY.

  33. 4.2 Some Ways to Prove Triangles Congruent Objectives • Learn about ways to prove triangles are congruent

  34. Don’t ASSume • Triangles cannot be assumed to be congruent because they “look” congruent. and • It’s not practical to cut them out and match them up so,

  35. We must show 6 congruent pairs • 3 angle pairs and • 3 pairs of sides

  36. WOW • That’s a lot of work Isn't there a shortcut ?

  37. Spaghetti Experiment • Using a small amount of playdough as your “points” put together a 5 inch, 3 inch and 2.5 inch piece of spaghetti to forma triangle. • Be careful, IT’S SPAGHETTI, and it will break.

  38. Compare your spaghetti triangle to your neighbors • Compare your spaghetti triangle to my spaghetti triangle. What do you notice ?

  39. We are lucky….. • There is a shortcut • We don’t have to show • ALL pairs of angles are congruent and • ALL pairs of sides are congruent

  40. If three sides of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. SSS Postulate E B C D A F

  41. Patty Paper Practice 5 inches 3 inches 2.5 inches

  42. Volunteer

  43. If two sides and the included angle are congruent to the corresponding parts of another triangle, then the triangles are congruent. SAS Postulate E B C D F

  44. If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. ASA Postulate E B C D A F

  45. The order of the letters MEAN something • Is SAS the same as SSA or A$$ ?

  46. Construction 2 Given an angle, construct a congruent angle. Given: Construct: Steps:

  47. Construction 3 Given an angle, construct the bisector of the angle Given: Construct: Steps:

  48. CAUTION !! Don't ASSume • If the diagram doesn’t show the markings or • You don’t have a reason • Shared sides, shared angles, vertical angles, parallel lines

  49. Remote Time Can the two triangles be proved congruent? If so, what postulate can be used? A. SSS Postulate • SAS Postulate • ASA Postulate • Cannot be proved congruent • I don’t know

  50. A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know