Section 1.4 If-Then Statements and Postulates

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Section 1.4 If-Then Statements and Postulates 4/6/2017 Geometry

Objectives-What we’ll learn Recognize and analyze a conditional statement Write postulates about points, lines, and planes using conditional statements 4/6/2017 Geometry

Postulate vs. Theorem A postulate is a statement that is assumed true without proof. A theorem is a true statement that can be proven.

Conditional Statement A conditional statement has two parts, a hypothesis and a conclusion. When conditional statements are written in if-then form, the part after the “if” is the hypothesis, and the part after the “then” is the conclusion. p → q represents “if p then q” 4/6/2017 Geometry

Examples If you are 13 years old, then you are a teenager. Hypothesis: Conclusion: You are a teenager 4/6/2017 Geometry

Rewrite in the if-then form (Conditional Statement) All mammals breathe oxygen If an animal is a mammal, then it breathes oxygen. A number divisible by 9 is also divisible by 3 If a number is divisible by 9, then it is divisible by 3. 4/6/2017 Geometry

Rewrite in the if-then form (Conditional Statement) Two lines intersect at a point. If two lines intersect, then they intersect at a point. Three non-collinear points determine a plane. If there are three non-collinear points, then they determine a plane. 4/6/2017 Geometry

Writing a Counterexample Write a counterexample to show that the following conditional statement is false If x2 = 16, then x = 4. As a counterexample, let x = -4. The hypothesis is true, but the conclusion is false. Therefore the conditional statement is false. 4/6/2017 Geometry

Converse The converse of a conditional statement is formed by switching the hypothesis and the conclusion. The converse of p → q is q → p 4/6/2017 Geometry

Rewrite in the Converse form. If you are 13 years old, then you are a teenager. If you are a teenager, then you are 13 years old. If a number divisible by 9, then it is also divisible by 3 If a number is divisible by 3, then it is divisible by 9.

Rewrite in the Converse form. If two angles are vertical angles, then they are congruent. If two angles are congruent, then they are vertical angles. If a quadrilateral has 4 right angles, then it is a rectangle. If a quadrilateral is a rectangle, then it has 4 right angles.

Point, Line, and Plane Postulates Postulate 1-1: Through any two points there exists exactly one line Postulate 1-2: Through any three noncollinear points there exists exactly one plane Postulate 1-3: A line contains at least two points Postulate 1-4: A plane contains at least three points not on the same line 4/6/2017 Geometry

Postulate 2-5: If two points lie in a plane, then the line containing them lies in the plane Postulate 2-6: If two planes intersect, then their intersection is a line 4/6/2017 Geometry