2 2 Analyze Conditional Statements n Definitions conditional

2. 2 Analyze Conditional Statements

n. Definitions: conditional statement: a logical statement that has two parts, a hypothesis and a conclusion. if-then form: a conditional statement written where “if” contains the hypothesis and “then” contains the conclusion. Example: On Sundays, there is football on TV. “If it is Sunday, then there is football on TV. ” hypothesis conclusion

Example 1 n. Rewrite the conditional statement in if-then form. (a) All 90° angles are right angles. If an angle measures 90°, then it is a right angle. (b) When n = 9, n 2 = 81. If n = 9, then n 2 = 81. (c) Three points are collinear if there is a line containing them. If there is a line containing three points, then the three points are collinear.

n. Definitions: Negation: the opposite of the original statement Statement: The ball is red. Negation: The ball is not red. Statement: The cat is not black. Negation: The cat is black.

Related Conditionals: Converse: exchange the hypothesis and conclusion Inverse: negate both the hypothesis and conclusion Contrapositive: exchange both the hypothesis and conclusion, then negate them both.

Example 2 Rewrite the conditional statement in if-then form, then write the converse, inverse, and contra-positive. “Since m A = 99º it is an obtuse angle. ” If-Then: If m A = 99º, then A an obtuse angle. Converse: If A is an obtuse angle, then m A = 99º. Inverse: If m A ≠ 99º, then A is not obtuse. Contrapositive: If A is not obtuse, then m A ≠ 99º. Which of the above statements are true?

You try! Rewrite the conditional statement in if-then form, then write the converse, inverse, and contra-positive. “The supplementary angles add up to 180º” If-Then: If two angles are complementary, then they add up to 180º Converse: If two angles add up to 180º, then they are supplementary. Inverse: If two angles are not supplementary, then they do Contrapositive: not add up to 180º If two angles do not add up to 180º, then they are not supplementary.

*With any definition, both the conditional statement and its converse are true. m Perpendicular lines: If two lines intersect to form a right angle, then they are perpendicular lines. Or. . . If two lines are perpendicular, then they intersect to form a right angle. l l⊥ m

Example 3 Determine whether the statement is a valid definition. a. If a polygon is a square, then the polygon has four congruent sides. b. If a polygon is both equilateral and equiangular, then the polygon is a regular polygon. c. If two angles have the same measure, then they are congruent.

Biconditional Statement: written when a conditional statement and its converse are true; contains the phrase “if and only if. ” m Definition: If two lines intersect to form a right angle, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form a right angle. Biconditional: Two lines are perpendicular if and only if they intersect to form a right angle. . *All definitions can be written as biconditional statements. l

Example 4 Write the definition of a right angle as (a) an if-then statement, (b) the converse of your if-then statement, and (c) a biconditional statement. (a) If an angle is a right angle, then the measure of the angle is 90º. (b) If the measure of an angle is 90º, then it is a right angle. (c) An angle is a right angle if and only if the measure of the angle is 90º.

Example 3 Determine whether the statement about the diagram is true. (a) AC ⊥ BD Yes. The right angle symbol indicates the lines intersect to form a right angle. (b) ∠AEB and ∠CEB form a linear pair. A Yes. The noncommon sides form a pair of opposite rays. (c) EA and EB are opposite rays. No. Point E does not lie on the same line as A and B, so the rays are not opposite. B C E D

Homework Pg 82 -84 #1, 3 -29, 39