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2.2 Definitions and Biconditional Statements Objectives: - Recognize and use definitions - Recognize and use biconditional statements
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A definition . . . <ul><li>Two lines are called perpendicular lines if the intersect to form a right angle. </li></ul><ul><li>A line perpendicular to a plane is a line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it. </li></ul><ul><li>The symbol is read as “is perpendicular to” </li></ul>
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Look at the picture on p. 79-bottom <ul><li>Using the definitions, you have learned, decide whether each statement is true: </li></ul><ul><li>Points D, X, and A are collinear. </li></ul><ul><li>Line AC is perpendicular to line DB </li></ul><ul><li>Angle AXB is adjacent to Angle CXD </li></ul>
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Using Biconditional Statements <ul><li>A biconditional statement is a statement that contains the phrase “if and only if.” </li></ul><ul><li>It is academic Friday only if I have tests in every class. </li></ul><ul><li>Three lines are coplanar if and only if they lie in the same plane. </li></ul>
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Rewriting a biconditional <ul><li>A biconditional statement can be rewritten as a conditional statement and its converse. </li></ul><ul><li>Biconditional: 3 lines are coplanar if and only if </li></ul><ul><li>they lie in the same plane. </li></ul><ul><li>Conditional: If 3 lines are coplanar, then they </li></ul><ul><li>lie in the same plane. </li></ul><ul><li>Converse: If 3 lines lie in the same plane, then </li></ul><ul><li>they are coplanar. </li></ul>
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A biconditional statement <ul><li>Can be either true or false. To be true, both the conditional statement and its converse must be true. This means that a true biconditional statement is true both forward and backward. All definitions can be written as true biconditional statements. </li></ul>
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X = 3 if and only if x 2 = 9 <ul><li>Is this a biconditional statement? </li></ul><ul><li>Yes, it contains “if and only if” </li></ul><ul><li>Is the statement true? </li></ul><ul><li>Yes, because the statement can be rewritten as: </li></ul><ul><li>Conditional: </li></ul><ul><li>If x = 3, then x 2 = 9 </li></ul><ul><li>Converse: </li></ul><ul><li>If x 2 = 9, then x = 3 </li></ul>
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Writing a Biconditional <ul><li>Is the converse true? If it is, combine it with the original to form a biconditional. If not, state a counterexample: </li></ul><ul><li>If two points lie in a plane, then the line containing them lies in the plane. </li></ul><ul><li>Converse: If a line containing 2 points lies in a plane, then </li></ul><ul><li>the points lie in the plane. </li></ul><ul><li>Biconditional: Two points lie in a plane if and only if </li></ul><ul><li>the line containing them lies in the plane. </li></ul>
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Writing a Biconditional <ul><li>Is the converse true? If it is, combine it with the original to form a biconditional. If not, state a counterexample: </li></ul><ul><li>If a number ends in 0, the the number is divisible by 5. </li></ul><ul><li>Converse: If a number is divisible by 5, then the number ends in zero. </li></ul><ul><li>This is false: 15 / 5 = 3 </li></ul>
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Writing a Postulate as a Biconditional <ul><li>Segment Addition Postulate: </li></ul><ul><ul><li>If B lies between points A and C, then AB + BC = AC </li></ul></ul><ul><li>Converse: </li></ul><ul><ul><li>If AB + BC = AC, then </li></ul></ul><ul><ul><li>B lies between A and C </li></ul></ul><ul><li>Biconditional: </li></ul><ul><ul><li>Point B lies between points A and C if and only if </li></ul></ul><ul><ul><li>AB + BC = AC </li></ul></ul>
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Do page 82 1 - 12 <ul><li>Homework: worksheets </li></ul>
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