1. Postulates and Algebraic Proofs Advanced Geometry Deductive Reasoning Lesson 2

2. Properties from Algebra Reflexive PropertyFor every number a, a = a. Symmetric PropertyFor all numbers a and b, if a = b, then b = a. Transitive PropertyFor all numbers a, b, and c, if a = b and b = c, then a = c. **A term is equal to itself.** ONE TERM **The two sides of an equation can be switched.** **Skip the middle term in the conclusion.** TWO TERMS THREE TERMS

3. Addition Propertyif a = b, then a + c = b + c. Subtraction Propertyif a = b, then a – c = b – c. Multiplication Propertyif a = b, then a c = b c. Division Property if a = b, then a ÷ c = b ÷ c. Properties from Algebra (cont.) You can add, subtract, multiply, or divide the same term on both sides of an equation. For all numbers a, b, and c,

4. Properties from Algebra (cont.) Substitution Property For all numbers a and b, if a = b, then a may be replaced by b in any equation or expression. Distributive Property For all numbers a, b and c, a(b + c) = ab + ac. Combine Like TermsTerms with like variables are combined without moving anything across the equal sign.

5. Examples: Name the property of equality that justifies each statement. If 3x + 7 = 12, then 3x = 5. If AB = CD, then AB + EF = CD + EF. If PQ + RS = 18 and RS = 8, then PQ + 8 = 18. Subtraction Property Addition Property Substitution Property

6. accepted to be true without proof Postulates Axiom is another word for postulate. Theorems has already been proven to be true can be used to justify that other statements are true

7. Through any two points there is exactly one line. The 1 st 7 Postulates Through any three points not on the same line there is exactly one plane. Two points determine a line. Three noncollinear points determine a plane.

8. A line contains at least two points. The 1 st 7 Postulates (cont.) A plane contains at least three points not on the same line.

9. If two planes intersect, then their intersection is a line. If two points lie in a plane, then the entire line containing those two points lies in that plane. If two lines intersect, then their intersection is exactly one point. The 1 st 7 Postulates (cont.)

10. Examples: Determine whether each statement is sometimes, always, or never true. If plane T contains and contains point G, then plane T contains point G. contains three noncollinear points. If and are coplanar, then they intersect. Always Never

11. Example: State the postulate that can be used to show each statement is true. B, D, and W are collinear. E, B, and R are coplanar.

12. Proof a logical argument Reasons: postulates (axioms) theorems definitions properties each statement made is supported by a reason

13. Given: M is the midpoint of Prove: From the definition of midpoint of a segment, PM = MQ. This means that and have the same measure. By the definition of congruence, if two segments have the same measure, then they are congruent. Thus, Paragraph Proof

14. Given: M is the midpoint of Prove: Two-Column Proof Proof: StatementsReasons a) M is the midpoint of a) Given b) PM = MQb) Definition of midpoint c) c) Definition of congruent

15. Solve for x. Show every step. 2x + 18 = 6

16. ALGEBRAIC PROOFS

17. Prove that if AB = CD, AB = 4x + 8 and CD = x + 2, then x = -2. Example: Write a complete proof for the situation below.

18. Prove that if, then. Example: Write a complete proof for the situation below.