Advanced Geometry Deductive Reasoning Lesson 2 Postulates and

Advanced Geometry Deductive Reasoning Lesson 2 Postulates and Algebraic Proofs

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

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

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 Terms with like variables are combined without moving anything across the equal sign.

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

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

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

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

The 1 st 7 Postulates (cont. ) • 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. • If two planes intersect, then their intersection is a line.

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. Always contains three noncollinear points. Never If and intersect. are coplanar, then they

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

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

Paragraph Proof 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,

Two-Column Proof Given: M is the midpoint of Prove: Proof: Statements Reasons a) M is the midpoint of a) Given b) PM = MQ b) Definition of midpoint c) c) Definition of congruent

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

ALGEBRAIC PROOFS

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

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