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

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

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

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

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

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

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. Included Angle Name the included angle: YE and ES ES and YS YS and YE

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

19. There is no such thing as an SSA postulate!
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. Included Side Name the included angle: Y and E E and S S and YYE 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. There is no such thing as an SSA postulate!
Warning: No SSA Postulate
There is no such thing as an SSA postulate! E B F A C D
NOT CONGRUENT

32. There is no such thing as an AAA postulate!
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 SAS SAS SSA SAS Vertical Angles Reflexive Property
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Vertical Angles
Reflexive Property
SSA
SAS

38. Name That Postulate
(when possible)

39. Name That Postulate
(when possible)

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