Postulates & Algebraic Proofs

1. Postulates & Algebraic Proofs Notes 13 - Sections 2.5 & 2.6

2. Essential Learnings • Students will understand and be able to identify and use basic postulates about points, lines, and planes. • Students will be able to use algebra to write two column proofs.

3. Vocabulary • Postulate or Axiom - a statement that is accepted as true without proof. • Proof - a logical argument in which each statement you make is supported by a statement that is accepted as true.

4. Vocabulary • Theorem - a statement or conjecture that has been proven. Can be used as a reason to justify statements in other proofs. • Paragraph Proof (informal proof) - involves writing a paragraph to explain why a conjecture for a given situation is true. • Algebraic proof – involves using a group of algebraic steps to solve problems and justify each step.

5. Postulate 2.1: Points, Lines & Planes • Through any two points, there is exactly one line. Example: Line n is the only line through points P and R.

6. Postulate 2.2 • Through any three noncollinear points, there is exactly one plane. Example: PlaneKis the only plane through noncollinear points A, B, and C.

7. Postulate 2.3 • A line contains at least two points. Example: Line n contains points P, Q and R.

8. Postulate 2.4 • A plane contains at least three noncollinear points. Example: PlaneKcontains noncollinear points L, B, C and E.

9. Postulate 2.5 • If two points lie in a plane, then the entire line containing those points lies in that plane. Example: Points A and B line in planeK, and line m contains points A and B, so line mis in planeK.

10. Postulate 2.6: Intersections of Lines & Planes • If two lines intersect, then their intersection is exactly one point. Example: Lines s and t intersect at point P.

11. Postulate 2.7 • If two planes intersect, then their intersection is a line. Example: PlanesFand G intersect in line w.

12. Example 1 • Explain how the picture illustrates that each statement is true. Then state the postulate that can be used to show each statement is true. a) Line m contains points F and G. Point E can also be on line m.

13. Example 1 cont. • Explain how the picture illustrates that each statement is true. Then state the postulate that can be used to show each statement is true. Lines s and t intersect at point D.

14. Example 2 • Determine whether each statement is always, sometimes, or never true. a) There is exactly one plane that contains points A, B, and C. b) Two lines intersect in two distinct points J and K.

15. Example 2 cont. • Determine whether each statement is always, sometimes, or never true. c) Points E and F are contained in exactly one line. d) A plane contains at least three lines.

16. Paragraph Proofs • To prove a conjecture, you use deductive reasoning to move from a hypothesis to the conclusion of the conjecture you are trying to prove. • Once a statement or conjecture has been proven, it is called a theorem and can be used to justify statements in proofs.

17. The Proof Process • 1. List the given information and, if possible, draw a diagram to illustrate the information. • 2. State the theorem or conjecture to be proven. • 3. Create a deductive argument by forming a logical chain of statements linking the given to what you are trying to prove. Given (hypothesis) Statements and Reasons Prove (Conclusion)

18. The Proof Process • 4. Justify each statement with a reason. Reasons include definitions, algebraic properties, postulates, and theorems. • 5. State what it is that you have proven. Given (hypothesis) Statements and Reasons Prove (Conclusion)

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

20. Properties of Real Numbers • The following properties are true for any real numbers a, b, and c. • Addition Property of Equality • Subtraction Property of Equality

21. Properties of Real Numbers • Multiplication Property of Equality • Division Property of Equality

22. Properties of Real Numbers • Reflexive Property of Equality • Symmetric Property of Equality • Transitive Property of Equality

23. Properties of Real Numbers • Substitution Property of Equality • Distributive Property of Equality

24. Example 3 - Algebraic Proof • Proof: a convincing argument which uses logic to show that something is true. • Given: 2(x - 3) = x + 1 Prove: x = 7 StatementReason

25. Assignment Page 128: 1-13 Page 135: 1-6, 13-17 (5, 6 & 17 are proofs) Math’s Mate Quiz – Friday