1. Warm-Up If m<J + m<E + m<R = 180°, then construct <R.

2. 4.4 Prove Triangles Congruent by SAS and HL 4.5 Prove Triangles Congruent by ASA and AAS Objectives: 1.To discover and use shortcuts for showing that two triangles are congruent

3. Congruent Triangles (CPCTC) congruent triangles cp ctc Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent. Corresponding sides are congruent Corresponding angles are congruent

4. 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!

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

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

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

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

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

10. Investigation: Shortcuts 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. We will test the remaining 5 in class. For each of these, you will be given three pieces to form a triangle. If the shortcut works, one and only one triangle can be made with those parts. Shortcuts?:  SSS  SSA  SAS  ASA  AAS  AAA √

11. Copying a Segment Draw segment AB on you paper. How could you copy that segment to another part of your paper using only a compass and a straightedge?

12. 1.Draw segment AB. Copying a Segment

13. 2.Draw a line with point A’ on one end. Copying a Segment

14. 3.Put point of compass on A and the pencil on B. Make a small arc. Copying a Segment

15. 4.Put point of compass on A’ and use the compass setting from Step 3 to make an arc that intersects the line. This is B’. Copying a Segment

16. Click on the image to watch a video of the construction. ©2001 New York State Regents Exam Prep Center

17. Copying an Angle Draw angle A on your paper. How could you copy that angle to another part of your paper using only a compass and a straightedge?

18. Copying an Angle 1.Draw angle A.

19. Copying an Angle 2.Draw a ray with endpoint A’.

20. Copying an Angle 3.Put point of compass on A and draw an arc that intersects both sides of the angle. Label these points B and C.

21. Copying an Angle 4.Put point of compass on A’ and use the compass setting from Step 3 to draw a similar arc on the ray. Label point B’ where the arc intersects the ray.

22. Copying an Angle 5.Put point of compass on B and pencil on C. Make a small arc.

23. Copying an Angle 6.Put point of compass on B’ and use the compass setting from Step 5 to draw an arc that intersects the arc from Step 4. Label the new point C’.

24. Copying an Angle 7.Draw ray A’C’.

25. Copying an Angle Click on the image to watch a video of the construction. ©2001 New York State Regents Exam Prep Center

26. Investigation: SAS Given the two sides and the included angle shown can you construct one and only one triangle ROW?

27. Investigation: ASA Given the two angles and the included side shown can you construct one and only one triangle SAM?

28. Investigation: AAS Given the two angles and the non- included side shown can you construct one and only one triangle JER? On this one, you’ll have to find the size of the third angle. How are you going to do that?

29. Investigation: SSA Given the two sides and the non-included angle shown can you construct one and only one triangle RAN? On this one, construct AN and angle N, then find a place for RA.

30. Investigation: AAA Given three angles shown can you construct one and only one triangle NAH? Pick a size for NA, and then copy the rest of the angles. I made it easy; the angles are all the same.

31. 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.

32. 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.

33. 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.

34. 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.

35. And One More! Hypotenuse-Leg (HL) Congruence Theorem: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

36. Example 1 What is the length of the missing leg in the each of the right triangles shown? Notice that the pieces given here correspond to SSA, which doesn’t work. Because of the Pythagorean Theorem, right triangles are an exception. Therefore, rt. triangles have theorems such as HL (hypotenuse-leg) and LL (leg-leg) 12

37. Example 2 Determine whether the triangles are congruent in each pair. Yes, SAS No

38. Example 3 Determine whether the triangles are congruent in each pair. Answer and explain which theorem in your notebook

39. Example 4 Explain the difference between the ASA and AAS congruence shortcuts. Answer in your notebook.

40. Example 5 TRY IT in your notebook! I will pick someone at random to work it on the board Ain’t life GRAND!

41. Assignment P. 243-246: 1, 2-4, 9-14, 16-18, 20-22, 26, 34, 35-38 (Pick One) P. 252-255: 2-7, 18-20, 28-30 (Pick one), 31-34 (Pick one), 36, 37