Section 1.7

1. Section 1.7 Deductive Structure

2. Objectives After studying this section, you will be able to: Recognize that geometry is based on a deductive structure Identify undefined terms, postulates, and definitions Understand the characteristics of theorems and the ways in which they can be used in proofs

3. Structure of Geometry Geometry is based on deductive structure A system of thought in which conclusions are justified by means of previously assumed or proved statements Every deductive structure contains the following 4 elements: Undefined terms Assumptions known as postulates Definitions Theorems and other conclusions

4. Undefined Terms, Postulates, and Definitions Postulate is an unproved assumption A definition states the meaning of a term or idea Definitions are reversible If a point is the midpoint of a segment, then the point divides the segment into two congruent segments If a point divides a segment into two congruent segments, then the point is the midpoint of the segment This is the form �if p then q�

5. If p then q p and q are declarative statements This is called a conditional statement or an implication The �if� part of the sentence is the called the hypothesis The �then� part of the sentence is called the conclusion �If p then q� can be symbolized as the following read as �p implies q� To write the converse of a conditional statement, you reverse parts p and q The converse of �if p then q� is �if q then p�

6. Theorems Theorems and postulates are not always reversible If two angles are right angles, then they are congruent is true If two angles are congruent, then they are right angles is false Definitions are always reversible, theorems and postulates are not always reversible

7. Practice Problems

8. Recap/Assignment What is a postulate? What is a definition? Which ones are reversible? Page 42 2-14 evens