1. Due MON 12/9 5.1 Indirect Proof p. 213 # 6-8,11-15 5.2 Proving That Lines are Parallel p. 219 # 10,12,15,19, 22-27

2. When disproving all options except for the one you want, you are doing an indirect proof. Since all of the other possibilities are incorrect, you are left with one correct option. Usually an indirect proof is used when proving things are not true. The steps are: A. List all possibilities for the conclusion. B. Assume that all possibilities you do not want to prove are true and use it as a given. (Assume that the negation of the desired conclusion is correct.) C. Write a chain of reasoning until you reach an impossibility. (A contradiction of: (a) the given information (b) a known fact (an already proved theorem, a definition, a postulate, etc.) D. State the remaining possibility as the desired conclusion.

3. A. List all possibilities for the conclusion. B. Assume that all possibilities you do not want to prove are true and use it as a given. (Assume that the negation of the desired conclusion is correct.)

4. StatementsReasons C. Write a chain of reasoning until you reach an impossibility. (A contradiction of: (a) the given information (b) a known fact (an already proved theorem, a definition, a postulate, etc.) D. State the remaining possibility as the desired conclusion.

5. What assumption should we Begin with?

6. Interior Angles Adjacent Interior Angle Exterior Angle Remote Interior Angle Exterior Angle Adjacent Interior Angle Remote Interior Angle Exterior Angle Adjacent Interior Angle Remote Interior Angle

7. Note: for the remainder of this presentation, the angle symbol ( ) will appear as . This is the Babylonio-Sumerian symbol for angle. No, not really. just a formatting bug I can’t figure out…

8. A B C D M A + m B + m ACB = 180  ACB and  ACD form a straight  180 = m  ACB + m  ACD m  A + m  B + m  ACB = m  ACB + m  ACD m  A + m  B = m  ACD m  ACD > m  A and m  ACD > m  B The measure of an exterior angle is greater than the measure of each remoter interior angle. The measure of an exterior angle equals the sum of the measures of the remoter interior angle.

9. m n t 6 4 If 2 lines are cut by a transversal such that 2 alternate interior angles are congruent, then the lines are parallel. Alt. int.  s  || lines If  4   6 then m || n.

10. m n t 6 4  4 is an ext. .  6 is a remote int.  4 >  6 This contradicts the given that  4 =  6  M || n

11. If 2 lines are cut by a transversal such that 2 alternate exterior angles are congruent, then the lines are parallel. Alt. ext.  s  || lines If  1   7 then m || n. m n t 1 7

12. m n t 1 7  1   7  3   1  7   5  3   5 m || n Alt. int.  s  || lines

13. If 2 lines are cut by a transversal such that 2 corresponding angles are congruent, then the lines are parallel. Corr.  s  || lines If  2   6 then m || n. m n t 2 6

14. m n t 2 6  2   6  4   2  4   6 m || n Alt. int.  s  || lines

15. If 2 lines are cut by a transversal such that 2 same side interior angles are supplementary, then the lines are parallel. Same side int.  s supp.  || lines If  4   5 then m || n. m n t 5 4

16. m n t 5 4  4 is supp.  5  4 and  3 form a straight   4 and  3 are supp.  5   3 m || n Alt. int.  s  || lines

17. m n t 1 8 If 2 lines are cut by a transversal such that 2 same side exterior angles are supplementary, then the lines are parallel. Same side ext.  s supp.  || lines If  1   8 then m || n.

18. m n t 1 8  1 is supp.  8  1 and  4 form a straight   1 and  4 are supp.  4   8 m || n corr.  s  || lines

19. If 2 coplanar lines are perpendicular to a third line, then they are parallel. Given: a  c and b  c Prove: a || b corr.  s  || lines