Note: Definitions are always reversible (converse is true) but postulates and theorems are not necessarily reversible.

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1 Honors Math 2 Deductive ing and Two-Column Proofs Name: Date: Deductive reasoning is a system of thought in which conclusions are justified by means of previously assumed or proven statements. Every deductive structure contains the following: Undefined terms (i.e., point, line, plane) Definitions See glossary of terms for some examples Postulates Conclusions that are accepted without proof Theorems mathematical statements that can be proved. Note: Definitions are always reversible (converse is true) but postulates and theorems are not necessarily reversible. Definition of Bisect If a ray divides an angle into two congruent angles, then Reverse: Postulates Properties of Equality and Properties of Congruence (See summary below) Two points determine a line. (Useful when adding line segments to a diagram) There exists one [median, angle bisector, midpoint, etc.] (Useful when adding to diagram) A whole equals the sum of its parts. Triangle congruence postulates and CPCTC (Corresponding parts of congruent triangles are congruent) Properties of Equality Addition Property If a = b and c = d, then a + c = b + d Subtraction Property If a = b and c = d, then a c = b d Multiplication Property If a = b, then ca = cb Division Property a If a = b and c 0, then = c Substitution Property If a = b then either a or b may be substituted for the other in any equation (or inequality) Reflexive Property a = a Symmetric Property If a = b and b = a Transitive Property If a = d and c = d, then a = c Distributive Property a ( b + c) = ab + ac b c

2 Many of the properties of equality can be extended into congruence in Geometry. Here are some examples: Properties of Congruence Addition Property If AB WX and CD YZ, then AB + CD WX +YZ Reflexive Property DE DE D D Substitution Property If 1 2 then either 1 or 2 may substituted for the other in any equation, congruency or statement Transitive Property DE FG and FG JK then DE JK How to Interpret a Diagram: You should Assume Straight lines and angles Collinearity of points Betweenness of points Relative position of points Given: Diagram as shown You Should Not Assume Right angles Congruent segments Congruent angles Relative sizes of segments and angles B Question: What should we assume? A C D E Question: What should we NOT assume? In a mathematical proof, every statement must be supported with a reason. Today, we will begin with some basic proofs and discover some reasons we already have at our disposal. Once you prove a theorem, you can use it as an assumption to prove other theorems.

3 Addition and Subtraction Properties Here are some ways you can use the Addition and Subtraction Properties of Congruence in proofs. If a segment (or angle) is added to congruent segments (or angles), then the sums are congruent. (Addition Property) If congruent segments (or angles) are added to congruent segments (or angles), then the sums are congruent. (Addition Property) If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) Given: AC DB Prove: AD CB 1. AC DB 1. Given 2. AD CB 2. Given: CBD EBA Prove: CBE DBA

4 Multiplication and Division Properties Here are some ways you can use the Multiplication and Division Properties of Congruence in proofs. If segments (or angles) are congruent, their like multiplies are congruent. (Multiplication Property) If segments (or angles) are congruent, their like divisions are congruent. (Division Property) Given: AB CD E is the midpoint of AB F is the midpoint of CD Prove: AE CF

5 Transitive Property Here are some ways you can use the Transitive Property of Congruence in proofs. If angles (or segments) are congruent to the same angle (or segment), then they are congruent to each other. If angles (or segments) are congruent to congruent angles (or segments), then they are congruent to each other. Given: DE AB EC AB Prove: AE bisects DC Reflexive Property An angle (or segment) is congruent to itself. Given: AD BC AB DC Prove: ABD CDB

6 Homework From textbook: p. 433 #14 and p #1, 2, 4, 5. In addition, complete the proofs below. Given: 1 4 Prove: 2 3 Given: BAD CAD AD bisects EAF Prove: BAE CAF