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Chapter 4

Chapter 4. Congruent Triangles. Sec . 4 – 1 Congruent Figures. Objective: 1) To recognize  figures & their corresponding parts. Congruent Polygons. Are the same size and the same shape. Fit exactly on top of each other Have  corresponding parts: Matching sides and s.

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Chapter 4

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  1. Chapter 4 Congruent Triangles

  2. Sec. 4 – 1Congruent Figures Objective: 1) To recognize  figures & their corresponding parts

  3. Congruent Polygons • Are the same size and the same shape. • Fit exactly on top of each other • Have  corresponding parts: • Matching sides and s

  4. Naming Polygons • List corresponding vertices in the same order. • Order Matters!! C B U T AB  ED  B  D  A  WU RS U S W A W R S E D

  5. Example: ΔWYS ΔMKV • mW = 25 • mY = 55 • Find mV K Y 55 M V 25 W 100 S

  6. Example 2: Congruence Statement Finish the following congruence statement: ΔJKL Δ_ _ _ M J ΔJKL  ΔNML L K N

  7. Theorem 4.1: • If two angles of one triangle are congruent to two angles of another, then the third angles are congruent

  8. Angles: A QB TC J Sides: AB QT BC TJ AC QJ Congruent Figures List the corresponding vertices in the same order. List the corresponding sides in the same order. 4-1

  9. Congruent Figures XYZKLM, mY = 67, and mM = 48. Find mX. Use the Triangle Angle-Sum Theorem and the definition of congruent polygons to find mX. mX + mY + mZ = 180 Triangle Angle-Sum Theorem mZ = mMCorresponding angles of congruent triangles that are congruent mZ = 48 Substitute 48 for mM. mX + 67 + 48 = 180 Substitute. mX + 115 = 180 Simplify. mX = 65 Subtract 115 from each side.

  10. If ABCCDE, then BACDCE. The diagram above shows BACDEC, not DCE. Corresponding angles are not necessarily congruent, therefore you cannot conclude that ABCCDE. Congruent Figures Can you conclude that ABCCDE in the figure below? List corresponding vertices in the same order. 4-1

  11. Congruent Polygons • Congruent polygons have congruent corresponding parts Name all corresponding parts.

  12. Write a congruence statement:

  13. GivenFind x, y, m, and n.

  14. Solve for x and y given:

  15. Given: Prove:

  16. 4.2 – Proving Triangles Congruent Using SSS and SAS HW : Objectives:

  17. Side Included Angle Side

  18. Side-Side-Side Postulate (SSS) If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

  19. Does the diagram give enough information to show that the triangles are congruent? Explain. Example 1 Use the SSS Congruence Postulate SOLUTION From the diagram you know that HJ LJ and HK  LK. By the Reflexive Property, you know that JK JK. Yes, enough information is given. Because corresponding sides are congruent, you can use the SSS Congruence Postulate to conclude that ∆HJK  ∆LJK. ANSWER

  20. Side-Angle-Side Postulate (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, then the two triangles are congruent.

  21. a. b. Example 2 Use the SAS Congruence Postulate Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning. SOLUTION From the diagram you know that AB CB and DB DB. a. The angle included between AB and DB is ABD. The angle included between CB and DB is CBD. Because the included angles are congruent, you can use the SAS Congruence Postulate to conclude that ∆ABD ∆CBD.

  22. Example 2 Use the SAS Congruence Postulate b. You know that GF GH and GE GE. However, the congruent angles are not included between the congruent sides, so you cannot use the SAS Congruence Postulate.

  23. 4.3 – Proving Triangles Congruent Using ASA

  24. Angle-Side-Angle Congruence Postulate 4-3

  25. Angle Side Angle Postulate (ASA) 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.

  26. Virtual Manipulatives - ASA

  27. Example 1 You are given that R Y and S X. b. You know that RTYZ,but these sides are not included between the congruent angles, so you cannot use the ASA Congruence Postulate. Determine When To Use ASA Congruence Based on the diagram, can you use the ASA Congruence Postulate to show that the triangles are congruent? Explain your reasoning. a. b. SOLUTION You are given that C E,B F, andBC FE. a. You can use the ASA Congruence Postulate to show that ∆ABC ∆DFE.

  28. Theorem 4-2

  29. 4.6 Hypotenuse leg

  30. Hypotenuse Leg Theorem 4-6 If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. The only version of SSA that is true!!

  31. Determine Whether Triangles are Congruent

  32. Are the triangles congruent? Explain

  33. SOME REASONS WE’LL BE USING • DEF OF MIDPOINT • DEF OF A BISECTOR • VERT ANGLES ARE CONGRUENT • DEF OF PERPENDICULAR BISECTOR • REFLEXIVE PROPERTY (COMMON SIDE) • PARALLEL LINES ….. ALT INT ANGLES

  34. SOLUTION The proof can be set up in two columns. The proof begins with the given information and ends with the statement you are trying to prove. Example 3 Write a Proof Write a two-column proof that shows ∆JKL  ∆NML. JL  NL Lis the midpoint of KM. ∆JKL  ∆NML

  35. Given:  and  . Write a proof to show that ∆DRA∆DRG. D A G R SOLUTION 1. Make a diagram and mark it up with the given information. Example 4 Prove Triangles are Congruent DR AG RA RG

  36. Checkpoint AC  AC  Statements Reasons _____ _____ _____ _____ ? ? ? ? DC CE CB  1. 1. Given ANSWER 2. 2. Given ANSWER DC Vertical Angles Theorem 3. BCA  ECD 3. ANSWER SAS Congruence Postulate 4. ∆BCA  ∆ECD 4. ANSWER Prove Triangles are Congruent 1. Fill in the missing statements and reasons. , ∆BCA  ∆ECD CB  CE

  37. Writing a Proof Using AAS

  38. More Proofs

  39. Given: Prove:

  40. Given: Prove:

  41. 4.4 Corresponding Parts of Congruent Triangles

  42. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) If ∆ABC ∆EDF Then CPCTC says:

  43. Prove: Given: Given Given Def of bisector Reflexive ASA CPCTC

  44. Given: Prove:

  45. Given: Prove:

  46. 4.5 Isosceles and Equilateral Triangles

  47. Isosceles Triangles Definition: a triangle with at least two congruent sides Leg Vertex Angle: The angle formed by the two congruent sides Leg Base Base angles: Formed by the base and one of the congruent sides

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