1. 4. 1 Apply Congruence and Triangles 4
4.1 Apply Congruence and Triangles 4.2 Prove Triangles Congruent by SSS, SAS
Objectives:
To define congruent triangles
To write a congruent statement
To prove triangles congruent by SSS, SAS

2. Congruent Polygons

3. Congruent Triangles (CPCTC)
Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent.

4. Congruence Statement
When naming two congruent triangles, order is very important.

5. Example
Which polygon is congruent to ABCDE? ABCDE  -?-

6. Properties of Congruent Triangles

7. Example
What is the relationship between C and F?

8. Third Angle Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

9. Congruent Triangles
Checking to see if 3 pairs of corresponding sides are congruent and then to see if 3 pairs of corresponding angles are congruent makes a total of SIX pairs of things, which is a lot! Surely there’s a shorter way!

10. Congruence Shortcuts?
Will one pair of congruent sides be sufficient? One pair of angles?

11. Congruence Shortcuts?
Will two congruent parts be sufficient?

12. Congruent Shortcuts? Will three congruent parts be sufficient?
And if so….what three parts?

13. Investigation…
… Using 2,3,4 Triangles

14. Side-Side-Side Congruence Postulate
SSS Congruence Postulate: If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

15. SSS Congruence Postulate

16. Using a 2-Column Proof! Is ABC is congruent to ABD? Why or why not?

17. Example
Decide whether the triangles are congruent. Explain your reasoning.

18. Investigation… …
Part 2

19. Congruence Shortcuts Side-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 another triangle, then the two triangles are congruent.

20. Can we prove?...yet?

21. What else would we need?

22. Can we Prove Triangle Congruence?

23. Another Proof…?

24. Congruent Shortcuts?
Will three congruent parts be sufficient?

25. Congruent Shortcuts? Will three congruent parts be sufficient?
Included Angle
Included Side

26. Congruent Shortcuts?
Will three congruent parts be sufficient?

27. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

28. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

29. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

30. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

31. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

32. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

33. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

34. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

35. Which case do we have? (SSS,SAS…) (They may not all work though!!!!)

36. Investigation: Shortcuts
SSS
AAS
SAS
AAA
ASA
ASS
Well, we know that SSS is a valid shortcut, and I’ll give you the hint that 2 others in the list do not work. To test the remaining 5, we will use our protractor and ruler. If the shortcut works, one and only one triangle can be made with those parts. √ √

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

38. Congruence Shortcuts
Side-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 another triangle, then the two triangles are congruent.

39. Congruence Shortcuts
Angle-Side-Angle (ASA) Congruence Postulate: 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.

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

41. Remember our justifications…
Vert angles
Linear Pair are Supp.
Alt Int Angles congruent
Defn of midpt
Defn of angle bisector
Right angles are congruent
Defn of 
…..

42. Classwork From 4.1 3 - 12 32 – 35 38 – 40 46 – 48* From 4.2 1 - 4
8 - 17
From 4.3
1 - 7 18 29