Reasoning & Proof

1. Reasoning & Proof Chapter 2

2. Postulates & Paragraph Proofs Lesson 2-5

3. Vocabulary • postulate • axiom • theorem • proof • paragraph proof • informal proof

4. Postulates • 2.1 – Through any two points, there is exactly one line. • 2-2 – Through any three points not on the same line, there is exactly one plane.

5. Example 1

6. Postulates 2.3 A line contains at least 2 points. 2.4 A plane contains at least 3 points not on the same line. 2.5 If two points lie in a plane, then the entire line containing those points lines in the plane. 2.6 If two lines intersect, then their intersection is exactly one point. 2.7 If two planes intersect, then their intersections is a line.

7. Example 2 Determine whether each statement is always, sometimes, or never true. Explain. • If points A, B, and C lie in plane M, then they are collinear. • There is exactly one plane that contains noncollinear points P, Q, and R. • There are at least two lines through points M and N.

8. Essential Parts of a Good Proof • State the given information. • State what is to be proven. • If possible, draw a diagram to illustrate the given information. • Develop a system of deductive reasoning.

9. Proof

10. Theorems 2.1 Midpoint Theorem - If M is the midpoint of AB, then AM ≅ MB.

11. Algebraic Proof Lesson 2-6

12. Vocabulary • Deductive argument • Two-column proof

13. Properties of Real Numbers

14. Example 1 Solve 3(x – 2) = 42. Justify each step.

15. Example 2

16. Example 3

17. Example 4

18. Proving Segment Relationships Lesson 2-7

19. Postulates 2.8 – Ruler Postulate – The points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero, and B corresponds to a positive real number. 2.9 – Segment Addition Postulate – If A, B, and C are collinear and B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

20. Proof

21. Theorems 2.2 – Segment Congruence – Congruence of segments is reflexive, symmetric, and transitive.

22. Proof

23. Proof

24. Proving Angle Relationships Lesson 2-8

25. Postulates 2.10 – Protractor Postulate – Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB such that the measure of the angle formed is r. 2.11 – Angle Addition Postulate – If R is in the interior of ∡PQS, then m∡PQR+ m∡RQS=m∡PQS. If m∡PQR+ m∡RQS=m∡PQS then R is in the interior of ∡PQS.

26. Example 1

27. Theorems 2.3 – Supplement Theorem – If two angels form a linear pair, then they are supplementary. 2.4 – Complement Theorem – If the non-common sides of two adjacent angles form a right angle, then the angles are complementary.

28. Example 2 If ∡1 and ∡2 form a linear pair, and m∡2 = 67, find m∡1.

29. Example 2 Find the measures of ∡3, ∡ 4, and ∡ 5 if m ∡ 3 = x + 20, m ∡ 4 = x + 40 and m ∡ 5 = x + 30.

30. Example 2 If ∡6 and ∡7 form a linear pair, and m∡6 = 3x + 32, m∡7 = 5x + 12 find x, m∡6, and m ∡7.

31. Theorems 2.5 Congruence of angles is reflexive, symmetric, and transitive.

32. Proof

33. Theorems 2.6 Angles supplementary to the same angle or to congruent angles are congruent. 2.7 Angles complementary to the same angle or to congruent angles are congruent.

34. Proof

35. Proof – Example 3

36. Example 4 If ∡1 and ∡2 are vertical angles and m ∡1 = x and m ∡2 = 228 – 3x, find m ∡1 and m ∡2.

37. Right Angle Theorems 2.9 – Perpendicular lines intersect to form four right angles. 2.10 – All right angles are congruent. 2.11 – Perpendicular lines form congruent adjacent angles. 2.12 – If two angles are congruent and supplementary, then each angle is a right angle. 2.13 – If two congruent angles form a linear pair, then they are right angles.