2.2 Definitions and Biconditional Statements GOAL 1 Recognize and use definitionsGOAL 2 Recognize and use biconditional statements. What you should learn.

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2.2 Definitions and Biconditional Statements GOAL 1 Recognize and use definitionsGOAL 2 Recognize and use biconditional statements. What you should learn You can use biconditional statements to help analyze geographic relations. Why you should learn it

GOAL 1 RECOGNIZING AND USING DEFINITIONS VOCABULARY perpendicular lines line perpendicular to a plane 2.2 Definitions and Biconditional Statements m n n P EXAMPLE 1

Extra Example 1 Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. RS U T a.Points R, S, and T are collinear. b. is perpendicular to. True; points are collinear if they lie on the same line. False; is not a right angle.

GOAL 2 USING BICONDITIONAL STATEMENTS VOCABULARY biconditional statement—contains the phrase “___________.” 2.2 Definitions and Biconditional Statements if and only if Writing a biconditional statement is the same as writing a conditional statement and its ________. converse EXAMPLE 2

Extra Example 2 EXAMPLE 3 Rewrite the biconditional statement as a conditional statement and its converse: Two lines intersect if and only if their intersection is exactly one point. Conditional statement: If two lines intersect, then they intersect in exactly one point. Converse: If two lines contain exactly one point, then the two lines intersect. Note: the hypothesis of the conditional comes first in the biconditional, and the conclusion follows “if and only if.”

Extra Example 3 Consider the following statement: x 2 < 49 if and only if x < 7. a. Is this a biconditional statement? b. Is the statement true? Yes; the statement contains “if and only if.” Rewrite the statement as a conditional and its converse: Conditional statement: Converse: If x 2 < 49, then x < 7. If x < 7, then x 2 < 49. The first statement is true, but the second one is false, so the biconditional statement is false.

Checkpoint Consider the following statement: x 2 = 4x if and only if x = 4. a.Is this a biconditional statement? b. Is the statement true? Yes. No. The conditional statement is not true; x 2 = 4x when x = 0.

WRITING A BICONDITIONAL STATEMENT 1.Write the converse of the statement and decide if it is true or false. 2.If it is true, combine it with the original statement to form a true biconditional. 3.If it is false, state a counterexample. EXAMPLE 4

Extra Example 4 The following statement is true. Write the converse and decide whether it is true or false. If the converse is true, combine it with the original to form a biconditional. If x 2 = 4, then x = 2 or x = -2. Converse:If x = 2 or x = -2, then x 2 = 4. True or false? TRUE Biconditional:x 2 = 4 if and only if x = 2 or x = -2. EXAMPLE 5

Extra Example 5 The converse of the Angle Addition Postulate is true. Write the converse and combine it with the postulate to form a true biconditional statement. Converse: then P is in the interior of Biconditional:P is in the interior of if and only if

Checkpoint The following statement is true. Write the converse and decide whether it is true or false. If the converse is true, combine it with the original to form a biconditional. If two planes intersect, then they contain the same line. Converse: If two planes contain the same line, then they intersect. True or False?TRUE Biconditional:Two planes intersect if and only if they contain the same line.

QUESTIONS?