Topic 2: Reasoning and Proof Pearson Unit 1 Topic 2: Reasoning and Proof 2-5: Reasoning in Algebra and Geometry Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: Foundation to TEKS (6) Use the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Reasoning: Proofs A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. An important part of writing a proof is giving justifications to show that every step is valid. A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs.

Copy the graphic organizer in the lower right When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. Hypothesis Conclusion Given info Definitions Postulates Properties Theorems Copy the graphic organizer in the lower right corner under the Distributive Property.

The Distributive Property states that a(b + c) = ab + ac. Remember!

Numbers are equal (=) and figures are congruent (). Remember!

Example 1: Write the correct PROPERTY. A  A Addition Property of Equality Reflexive Property of Congruence Division Property of Equality If AB  CD and CD  EF, then AB  EF. Transitive Property of Equality Multiplication Property of Equality Transitive Property of Congruence Distributive Property Subtraction Property of Equality Substitution Property of Equality

More practice (not in your notes) Identify the property that justifies each statement. QRS  QRS B. m1 = m2 so m2 = m1 C. AB  CD and CD  EF, so AB  EF. D. 32° = 32° Reflexive Property of  Symmetric Property of = Transitive Property of  Reflexive Property of =

A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column.

Example: 2 Statement Justification 1) AB bisects  RAN 1) Given 2) m RAB = m BAN 2) Definition of angle bisector 3) x = 2x - 75 3) Substitution Property of Equality 4) -x = -75 4) Subtraction Property of Equality 5) x = 75 5) Division Property of Equality

Midpoint: The point that bisects a segment. DEFINITION: Midpoint: The point that bisects a segment. A M B

Example: 3 Write a justification for each step, given that A and B are supplementary and mA = 45°. Statements Reasons 1. A and B are supplementary. mA = 45° 1. Given 2. mA + mB = 180° 2. Def. of supplementary s 3. 45° + mB = 180° 3. Substitution Property of = 4. mB = 135° 4. Subtraction Property of =

Example: 4 Write a justification for each step, given that B is the midpoint of AC and AB  EF. Statements Reasons 1. B is the midpoint of AC. 1. Given 2. AB  BC 2. Def. of midpoint 3. Given 3. AB  EF 4. Transitive Prop. of  4. BC  EF

Example 5: Statement Justification 1) AB  CD 1) Given AB = CD 2) Definition of congruent segments 3) AB + BC = CD + BC 3) Addition Property of Equality 4) AC = AB + BC BD = CD + BC 4) Segment Addition Postulate 5) AC = BD 5) Substitution Property of Equality 6) AC  BD 6) Definition of congruent segments