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Congruent Triangles

Congruent Triangles. Polygons MNOL and ZYXW are congruent. ∆ABC and ∆DEF are congruent. Rectangles ABCD and EFGH are not congruent. A. D. Y. Z. E. H. ∆ZXY and ∆JLP are not congruent. L. X. B. C. G. F. J. P. 4 -1 Congruent Figures. Objective:

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Congruent Triangles

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  1. Congruent Triangles

  2. Polygons MNOL and ZYXW are congruent ∆ABC and ∆DEF are congruent Rectangles ABCD and EFGH are not congruent A D Y Z E H ∆ZXY and ∆JLP are not congruent L X B C G F J P

  3. 4-1 Congruent Figures Objective: To recognize congruent figures and their corresponding parts

  4. Vocabulary/ Key Concept • Congruent polygons- two polygons are congruent if their corresponding sides and angles are congruent

  5. Naming Congruent Figures Ang Legs Triangle: Construct two triangles with the following sides-1 red, 1 blue, 1 yellow ∆ABC and ∆DEF óA óB óC

  6. Warm Up: WXYZ  JKLM. List 4 pairs of congruent sides and angles. • WX  JK • XY  KL • YZ  LM • ZW MJ • W  J • K  X • Y  L • Z  M

  7. Each pair of polygons are congruent. Find the measure of each numbered angle • M1 = 110 • m 2 = 120 • M3 = 90 • m 4 = 135

  8. We know: óBóF óAóE Then we can conclude: óCóD Key Concept: If two angles in a triangle are congruent to two angles in another triangle, then the third angles are congruent. WARNING:This is only true for ANGLES not side lengths!

  9. Concept Check! How do we know if two triangles are congruent?

  10. Objective: To prove two triangles are congruent using SSS and SAS Postulates

  11. Key Concepts • SSS – Side-side-side corresponding congruence. If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent (all corresponding sides are equal)

  12. Example 1: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent. Student Slide #1

  13. Example 2: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent. Student Slide #2

  14. Key Concepts • SAS – Side-Angle-Side corresponding Congruence. ANGLE MUST BE IN BETWEEN THE TWO SIDES (INCLUDED ANGLE) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent

  15. Example 1: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent. Student Slide #3

  16. Example 2: Can you use SAS to prove these two triangles are congruent? If no, what information would you need in order to use SAS to prove these triangles are congruent? Student Slide #4

  17. Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement. AB  CB --CONGRUENCE MARKING BD  BD – REFLEXIVE PROPERTY OF CONGRUENCE ABD  CBD–CONGRUENCE MARKING  ABD   CBD by SAS

  18. Example: If we know: óBóE What other information must we know in order to prove ∆ABC ∆DEF using SAS?

  19. WARM UP (will be collected): Name the three pairs of corresponding sides Name the three pairs of corresponding angles Do we have enough information to conclude that the two triangles are congruent? Explain your reasoning. *CORRESPONDING DOES NOT MEAN THEY ARE CONGRUENT!

  20. WUP#1: Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement. What do you know? NP QP -- CONGRUENT MARKS NR QR -- CONGRUENT MARKS RP RP -- REFLEXIVE PROPERTY OF   PRN  PRQ by SSS

  21. WUP #2: What one piece of additional information must we know in order to prove the triangles are congruent using SAS. Explain your reasoning and then write a congruence statement. Explanation: Statement:

  22. Objective: To prove two triangles are congruent using ASA, AAS, and HL Postulates

  23. Key Concepts • ASA – Two angles and an included side. SIDE IS IN BETWEEN THE ANGLES If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

  24. Key Concepts • AAS – Two angles and a non-included side. If two angles and the non-included side of a triangle are congruent to two angles and the non-included side of another triangle, then the two triangles are congruent.

  25. Determine if you can use ASA or AAS to prove two triangles are congruent. Write the congruence statement.

  26. Determine if you can use ASA or AAS to prove two triangles are congruent and explain your reasoning. Then write the congruence statement. Explain:

  27. Determine if you can use ASA or AAS to prove two triangles are congruent and explain your reasoning. Then write the congruence statement. Explain: TRY ONE

  28. Congruence that works:  Congruence that does not work:  ASS SSA AAA SSS SAS AAS ASA *Remember, we don’t swear in math (not even backwards). And no screaming!

  29. What did you learn today? • What are the five ways (one for right triangles) to prove triangles are congruent?

  30. Example 1: Complete the 2 column proof: Given: óBAEóEDB, óABEóDEB Prove: óABEóDEB Statements Reasons

  31. So what do we know about the parts of congruent triangles? Corresponding Parts of Congruent Triangles are Congruent CPCTC Hence, *Remember, you can only use CPCTC, AFTER you have proven two triangles to be congruent!

  32. Write a Proof Statement • FJ  GHJFH  GHF • HF  FH •  JFH  GHF • FG  JH Reasons • Given • Reflexive property of congruence • SAS • CPCTC

  33. TRY ONE: Write a Proof Given: óBAC óCDE, AC  CD Prove: óB óE Statement • AC  CD, óBAC óCDE •  ACB  ECD •  DEC   ABC •  B  E Reasons • Given • Vertical angles • ASA • CPCTC

  34. What did you learn today? • What does CPCTC mean and when do we use it?

  35. CPCTC Song (sung to the tune of “YMCA” by the Village People) Author of lyrics: Eagler Young man, there's no need to feel down I said, young man, pick yourself off the ground I said, young man, 'cause there's a new thing I've found There's no need to be unhappy Young man, there's this thing you can do I said, young man, it's so easy to prove You can use it, and I'm sure you will see Many ways to show congruency It's fun to solve it with C-P-C-T-C It's fun to solve it with C-P-C-T-C Barely takes any time, uses only one line It's the easiest thing you'll find It's fun to solve it with C-P-C-T-C It's fun to solve it with C-P-C-T-C If you don't have a clue, it's so simple to do Write five letters and you'll be through

  36. Mini Lab

  37. Similar Polygons: Two polygons are similar if: Corresponding angles are congruent Corresponding sides are proportional

  38. Determine whether rectangle HJKL is similar to rectangle MNPQ.

  39. Transformations

  40. Reflection over the x-axis Preserves congruence

  41. Rotation 90ô about the origin 90ô (x,y)  (-y,x) Ex: D(6,3)D’(-3,6) 180ô (x,y)(-x,-y) Preserves congruence

  42. Translation 3 units right, 4 units down Preserves congruence

  43. Dilation Scale factor (length of image/length of original) :2/1 Does not preserve congruence BUT, are they similar?

  44. Description:

  45. Description:

  46. 1) Which transformations preserve congruence? 2) What criteria needs to be met for triangles to be similar?

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