2.2 Analyze Conditional Statements

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2.2 Analyze Conditional Statements Geometry 2.2 Analyze Conditional Statements

Using Conditional Statements Logic = Reasoning Logical statements can be written as a conditional statement also known as an If-Then statement. Symbolically it is written: If (hypothesis) , then (conclusion). If an angle has a measure of 90o, then the angle is a right angle.

Parts of a conditional statement A conditional statement can be broken up into two parts. The statement after the If is called the hypothesis and the statement after the then is called the Conclusion. If an angle has a measure of 90o, then the angle is a right angle. What is the hypothesis? What is the conclusion?

Negation: p p= p= q= q= The negation of a hypothesis or conclusion is created by “denying” the original hypothesis or conclusion. For example, given: “If I clean my room, then I can go to the movies”. p= p= q= q=

Truths and Counterexamples Conditional statements can be true or false. A conditional statement is true if the conclusion is true whenever the hypothesis is true. A conditional statement is true only if a counterexample (one example that contradicts the conditional statement) can not be found to disprove the statement. If the keys are in the ignition, then the car is running. Is this statement true?

The Converse If p, then q. were p is the hypothesis and q is the conclusion. The converse is formed by switching the hypothesis and the conclusion.

The Converse (cont’d) If an angle has a measure of 90o, then the angle is a right angle. If the angle is a right angle, then the angle has a measure of 90o. Example: If I practice well, then I will play Hypothesis: Conclusion: Converse:

Contrapositive: q  p = The contrapositive of a conditional statement is created by negating the converse of the original conditional statement. “If I clean my room, then I can go to the movies”. qp = q  p =

Contrapositive The contrapositive is true only if the original conditional statement is true. The converse truth or falsity has no bearing on the truth or falsity of the original conditional statement

Conditional Statement, Converse, Contrapositives Given: “If I do my homework, then I will pass” p= q= p= q= q  p

More Examples: p= q= q  p Given: “If I don’t go shopping, then I will run out of milk.” p= q= p= q= q  p

More Examples p= q= q  p Given: “If I don’t turn on the heat, I will get cold.” p= q= p= q= q  p

Rules for providing proof Definitions Postulates Theorems Axioms Properties Constructions

Biconditional Statement A biconditional statement or “p if and only if q” is written when a conditional statement AND its converse is always true. If A is a right angle, then its measure is 90o. If A has a measure of 90o, then it is a right angle. A is a right angle if and only if its measure is 90o.