Proving Triangles Congruent

1. Proving Triangles Congruent Powerpoint hosted on www.worldofteaching.com Please visit for 100’s more free powerpoints

2. Why do we study triangles so much? • Because they are the only “rigid” shape. • Show pictures of triangles

3. F B A C E D The Idea of a Congruence Two geometric figures with exactly the same size and shape.

4. How much do you need to know. . . . . . about two triangles to prove that they are congruent?

5. Corresponding Parts • AB DE • BC EF • AC DF •  A  D •  B  E •  C  F B A C E F D If all six pairs of corresponding parts of two triangles (sides and angles) are congruent, then the triangles are congruent. ABC DEF

6. SSS SAS ASA AAS Do you need all six ? NO !

7. Side-Side-Side (SSS) E B F A D C • AB DE • BC EF • AC DF ABC DEF Order matters!!

8. Distance Formula • The Cartesian coordinate system is really two number lines that are perpendicular to each other. • We can find distance between two points via……….. • Mr. Moss’s favorite formula a2 + b2 = c2

9. Distance Formula • The distance formula is from Pythagorean’s Theorem. • Remember distance on the number line is the difference of the two numbers, so • And • So

10. To find distance: • You can use the Pythagorean theorem or the distance formula. • You can get the lengths of the legs by subtracting or counting. • A good method is to align the points vertically and then subtract them to get the distance between them

11. Find the distance between (-3, 7) and (4, 3) _____

12. Do some examples via Geo Sketch

13. Class Work • Page 240, # 1 – 13 all

14. Warm Up • Are these triangle pairs congruent by SSS? Give congruence statement or reason why not.

15. Side-Angle-Side (SAS) B E F A C D • AB DE • A D • AC DF ABC DEF included angle

16. Included Angle The angle between two sides H G I

17. E Y S Included Angle Name the included angle: YE and ES ES and YS YS and YE E S Y

18. Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C F D NOT CONGRUENT

19. Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT

20. Why no SSA Postulate? • We would get in trouble if someone spelled it backwards. (bummer!) • Really – SSA allows the possibility of having two different solutions. • Remember: It only takes one counter example to prove a conjecture wrong. • Show problem on Geo Sketch

21. * A Right Δ has one Right . Hypotenuse – - Longest side of a Rt. Δ - Opp. of Right  Leg Leg – one of the sides that form the right  of the Δ. Shorter than the hyp. Right 

22. Th(4-6) Hyp – Leg Thm. (HL) • If the hyp. & a leg of 1 right Δ are  to the hyp & a leg of another rt. Δ, then the Δ’s are .

23. Group Class Work • Do problem 18 on page 245

24. Class Work • Pg 244 # 1 – 17 all

25. Warm Up • Are these triangle pairs congruent? State postulate or theorem and congruence statement or reason why not.

26. Angle-Side-Angle (ASA) B E F A C D • A D • AB DE • B E ABC DEF included side

27. Included Side The side between two angles GI GH HI

28. E Y S Included Side Name the included angle: Y and E E and S S and Y YE ES SY

29. Angle-Angle-Side (AAS) B E F A C D • A D • B E • BC  EF ABC DEF Non-included side

30. Angle-Angle-Side (AAS) • The proof of this is based on ASA. • If you know two angles, you really know all three angles. • The difference between ASA and AAS is just the location of the angles and side. • ASA – Side is between the angles • AAS –Side is not between the angles

31. Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT

32. Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C F D NOT CONGRUENT

33. Triangle Congruence (5 of them) • SSS postulate • ASA postulate • SAS postulate • AAS theorem • HL Theorem

34. Right Triangle Congruence When dealing with right triangles, you will sometimes see the following definitions of congruence: HL - Hypotenuse Leg HA – Hypotenuse Angle (AAS) LA - Leg Angle (AAS of ASA) LL – Leg Leg Theorem (SAS)

35. Name That Postulate (when possible) SAS ASA SSA SSS

36. Name That Postulate (when possible) AAA ASA SSA SAS

37. Name That Postulate (when possible) Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA SAS

38. Name That Postulate (when possible)

39. Name That Postulate (when possible)

40. Let’s Practice ACFE Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AF For AAS:

41. Review Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS:

42. Class Work • Pg 251, # 1 – 17 all

43. CPCTC • Proving shapes are congruent proves ALL corresponding dimensions are congruent. • For Triangles: Corresponding Parts of Congruent Triangles are Congruent • This includes, but not limited to, corresponding sides, angles, altitudes, medians, centroids, incenters, circumcenters, etc. They are ALL CONGRUENT!

44. Warm Up