Conditional & Biconditional Statements Chapter 2 Section 4

HypothesisConclusion IF then Examples: If it is a nice day then I will go to the park Conditional Statements an “if-then” statement Symbolic Notation p  q (If p then q) q  p (If q then p)

Converse The converse of a conditional switches the hypothesis and the conclusion. Example Conditional: If 2 lines intersect to form right angles then they are perpendicular. Converse: If 2 lines are perpendicular then they intersect to form right angles.

Truth Value A conditional statement can have a truth value of true or false. To show a conditional is TRUE Show that every time the hypothesis is true, the conclusion is true To show a conditional is FALSE Find a counterexample in which the hypothesis is TRUE and the conclusion is FALSE

Write the converse of the conditional statement. a) If an angle has less than 90 degrees, then it is an acute angle. If an angle is acute, then it measures less than 90 degrees. If a figure is a square, then it has four congruent sides. b) If a figure has four congruent sides, then it is a square. Practice/ Review

Biconditional Statement the combination of a conditional statement and its converse (as long as both statements are true). statements are combined using the phrase “if and only if” Example: Two angles have the same measure if and only if the angles are congruent.

Symbolic Form StatementExampleSymbolic FormRead Conditional If there is lightning, then there is thunder. p→qp→qIf p, then q Converse If there is thunder, then there is lightning. q→pq→pIf q, then p Biconditional There is lightning if and only if there is thunder. p↔qp↔qp iff q

Biconditional: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Conditional: If a number is divisible by 3, then the sum of its digits is divisible by 3. Conditional (converse): If the sum of a number’s digits is divisible by 3, then the number is divisible by 3. Separate the biconditional into two conditional statements. Example 1

Biconditional: Two angles are congruent if and only if they have the same measure. Conditional: If two angles are congruent, then they have the same measure. Conditional (converse): If two angles have the same measure, then they are congruent. Separate the biconditional into two conditional statements. Example 2

Conditional: If three points are collinear, then they lie on the same line. Converse: If three points lie on the same line, then they are collinear. TRUE Biconditional: Three points are collinear if and only if they lie on the same line. Example 3 Write the biconditional:

Conditional: A square has a side length of 5 if and only if it has an area of 25. Converse: If a square has an area of 25, then it has a side length of 5. TRUE Biconditional: A square has a side length of 5 if and only if it has an area of 25. Example 4 Write the biconditional:

Writing Definitions as Biconditional Statements: Uses clearly understandable terms Precise. (avoids words such as “large, sort of, and some”) Reversable. (must be able to be written as a TRUE converse). A Good Definition:

A triangle is a three-sided polygon. Biconditional: The triangle is a polygon if and only if it is a three- sided polygon. Example 5 Write the definition as a biconditional: The measure of a straight angle is 180 degrees. Biconditional: The measure if a straight angle if and only if it is 180 degrees.

Assignment #20 Pg.99 #1-19 all