Proving Triangles are Congruent SSS, SAS; ASA; AAS

1. Proving Triangles are CongruentSSS, SAS; ASA; AAS CCSS: G.CO7

2. Standards for Mathematical Practices • 1. Make sense of problems and persevere in solving them. • 2. Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others.   • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6. Attend to precision. • 7. Look for and make use of structure. • 8. Look for and express regularity in repeated reasoning.

3. CCSS:G.CO 7 • USE the definition of congruence in terms of rigid motions to SHOW that two triangles ARE congruent if and only if corresponding pairs of sides and corresponding pairs of angles ARE congruent.

4. ESSENTIAL QUESTION • How do we show that triangles are congruent? • How do we use triangle congruence to plane and write proves ,and prove that constructions are valid?

5. Objectives: • Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem • Use congruence postulates and theorems in real-life problems.

6. Proving Triangles are Congruent: SSS and SAS

7. SSS AND SASCONGRUENCE POSTULATES then If 1.ABDE 4.AD 2.BCEF 5. BE ABCDEF 3.ACDF 6.CF If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Sides are congruent Angles are congruent Triangles are congruent and

8. SSS AND SASCONGRUENCE POSTULATES S S S Side MNQR then MNPQRS Side NPRS Side PMSQ POSTULATE POSTULATE 19Side -Side -Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent. If

9. SSS AND SASCONGRUENCE POSTULATES The SSS Congruence Postulate is a shortcut for provingtwo triangles are congruent without using all six pairsof corresponding parts.

10. SSS AND SASCONGRUENCE POSTULATES POSTULATE Side PQWX A S S then PQSWXY Angle QX Side QSXY POSTULATE 20Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If

11. Congruent Triangles in a Coordinate Plane AC FH ABFG Use the SSS Congruence Postulate to show that ABCFGH. SOLUTION AC = 3 and FH= 3 AB = 5 and FG= 5

12. Congruent Triangles in a Coordinate Plane d = (x2 – x1 )2+ (y2 – y1 )2 d = (x2 – x1 )2+ (y2 – y1 )2 BC = (–4 – (–7))2+ (5– 0)2 GH = (6 – 1)2+ (5– 2)2 = 32+ 52 = 52+ 32 = 34 = 34 Use the distance formula to find lengths BC and GH.

13. Congruent Triangles in a Coordinate Plane BCGH BC = 34 and GH= 34 All three pairs of corresponding sides are congruent, ABCFGH by the SSS Congruence Postulate.

14. SAS  postulate SSS  postulate

15. T C S G The vertex of the included angle is the point in common. SSS  postulate SAS  postulate

16. SSS  postulate Not enough info

17. SSS  postulate SAS  postulate

18. Not Enough Info SAS  postulate

19. SSS  postulate Not Enough Info

20. SAS  postulate SAS  postulate

22. Congruent Triangles in a Coordinate Plane MN DE PMFE Use the SSS Congruence Postulate to show that NMPDEF. SOLUTION MN = 4 and DE= 4 PM = 5 and FE= 5

23. Congruent Triangles in a Coordinate Plane d = (x2 – x1 )2+ (y2 – y1 )2 d = (x2 – x1 )2+ (y2 – y1 )2 PN = (–1 – (– 5))2+ (6– 1)2 FD = (2 – 6)2+ (6– 1)2 = 42+ 52 = (-4)2+ 52 = 41 = 41 Use the distance formula to find lengths PN and FD.

24. Congruent Triangles in a Coordinate Plane PNFD PN = 41 and FD= 41 All three pairs of corresponding sides are congruent, NMPDEF by the SSS Congruence Postulate.

25. Proving Triangles are Congruent ASA; AAS

26. If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate

27. If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

28. Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

29. You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem

30. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. Ex. 1 Developing Proof

31. A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG. Ex. 1 Developing Proof

32. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. Ex. 1 Developing Proof

33. B. In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent. Ex. 1 Developing Proof

34. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. UZ ║WX AND UW ║WX. Ex. 1 Developing Proof 1 2 3 4

35. The two pairs of parallel sides can be used to show 1  3 and 2  4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate. Ex. 1 Developing Proof 1 2 3 4

36. Given: AD ║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC . Use the fact that AD ║EC to identify a pair of congruent angles. Ex. 2 Proving Triangles are Congruent

37. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1. Proof:

38. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1. Given Proof:

39. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Proof:

40. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles Proof:

41. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles Vertical Angles Theorem Proof:

42. Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Given Alternate Interior Angles Vertical Angles Theorem ASA Congruence Theorem Proof:

43. Note: • You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.