Tuesday, September 17 th

1. Tuesday, September 17th Warm Up Fill in the Proofs 2. Fill in the proof Solve for 2x + 4 = 12 Solve for ½ x = 12

2. Important Next Quiz: Friday 9/20 PROOFS

3. Geometric Proofs

4. Definitions • Postulates • Properties • Theorems Conclusion Hypothesis When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them.

5. A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs.

9. #1 Write a justification for each step, given that A and Bare supplementary and mA = 45°. 1. Aand Bare supplementary. mA = 45° Given information Def. of supp s 2. mA+ mB= 180° Subst. Prop of = 3. 45°+ mB= 180° Steps 1, 2 Subtr. Prop of = 4. mB= 135°

10. #2 Use the given plan to write a two-column proof. Given: 1 and 2 are supplementary, and 1  3 Prove: 3 and 2 are supplementary. Plan: Use the definitions of supplementary and congruent angles and substitution to show that m3 + m2 = 180°.By the definition of supplementary angles, 3 and 2are supplementary.

11. Example 2 Continued Given 1 and 2 are supplementary. 1  3 m1+ m2 = 180° Def. of supp. s m1= m3 Def. of s Subst. m3+ m2 = 180° Def. of supp. s 3 and 2 are supplementary

12. #3 Write a justification for each step, given that mABC= 90° and m1= 4m2. 1. mABC= 90° and m1= 4m2 2. m1+ m2 = mABC 3. 4m2 + m2 = 90° 4. 5m2= 90° 5. m2= 18° Given  Add. Post. Subst. Simplify Div. Prop. of =.

13. #4 2. Use the given plan to write a two-column proof. Given: 1, 2 , 3, 4 Prove: m1 + m2 = m1 + m4 Plan: Use the linear Pair Theorem to show that the angle pairs are supplementary. Then use the definition of supplementary and substitution. 1. 1 and 2 are supp. 1 and 4 are supp. 1. Linear Pair Thm. 2. Def. of supp. s 2. m1+ m2 = 180°, m1+ m4 = 180° 3. Subst. 3. m1+ m2 = m1+ m4

14. CW Justification Card Practice #1 Answers

15. Homework Answers

16. Congruent Triangles Congruent triangles have 3 congruent sides and 3 congruent angles. The parts of congruent triangles that “match” are called corresponding parts.

17. Congruence Statement In a congruence statement ORDER MATTERS!!!! Everything matches up.

18. CPCTC Corresponding Parts of Congruent Triangles are Congruent

19. Complete each congruence statement. B If ABC  DEF, then BC  ___ A C D F E

20. Complete each congruence statement. B If ABC  DEF, then A  ___ A C D F E

21. Complete each congruence statement. B If ABC  DEF, then C  ___ F A C D F E

22. Fill in the blanks If CAT  DOG, then AC  ___

23. Fill in the blanks BAT  MON T  ___ _____  ONM _____  MO NM  ____

24. Fill in the blanks BCA   ____ ____   GFE

25. Complete the congruence statement. _____   JKN

26. Complete the congruence statement. _____   CBD

27. There are 5 ways to prove triangles congruent.

28. We Use • Sides • Angles

29. Side-Side-Side (SSS) Congruence Postulate All Three sides in one triangle are congruent to all three sides in the other triangle

30. Side-Angle-Side (SAS) Congruence Postulate Two sides and the INCLUDED angle (the angle is in between the 2 marked sides)

31. A A A A S S Angle-Angle-Side (AAS) Congruence Postulate Two Angles and One Side that is NOT included

32. Angle-Side-Angle (ASA) Congruence Postulate A A S S A A Two angles and the INCLUDED side (the side is in between the 2 marked angles)

33. There is one more way to prove triangles congruent, but it’s only for RIGHT TRIANGLES…Hypotenuse Leg HL

34. SSS SAS ASA AAS HL NO BAD WORDS The ONLY Ways To Prove Triangles Are Congruent

35. 2 markings you can add if they aren’t marked already

36. Share a side Reason: reflexive property Vertical Angles Reason: Vertical Angles are congruent

37. Practice 1-3

38. Homework Finish Triangle Congruence Notes