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Section 2-2

Section 2-2. Biconditional Statements. Biconditional statement. a statement that contains the phrase “if and only if”. Equivalent to a conditional statement and its converse. We can use iff to stand for “If and only if”.

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Section 2-2

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  1. Section 2-2 Biconditional Statements

  2. Biconditional statement • a statement that contains the phrase “if and only if”. • Equivalent to a conditional statement and its converse.

  3. We can use iff to stand for “If and only if”

  4. In order for a biconditional statement to be TRUE, both the conditional statement and its converse must be true.

  5. Two lines intersect if and only if their intersection is exactly one point. Example #1: Write this biconditional statement as a conditional statement.

  6. Conditional Statement: • If two lines intersect, then • their intersection is exactly • one point. True

  7. Now write the converse. • If their intersection is exactly one point, then two lines intersect. True

  8. Example #2 Write this biconditional statement as a conditional statement. • Three lines are coplanar if and only if they lie in the same plane.

  9. If three lines are coplanar, then they lie in the same plane. Conditional Statement: True

  10. If three lines lie in the same plane, then they are coplanar. Now write the converse. True

  11. Write the conditional as a biconditional statement. • If an angle is acute then it has a measure between 0° and 90°.

  12. Write the converse • If an angle has a measure between 0° and 90°, then it is acute. True

  13. Identify whether the converse is true or false • If it is true, then a biconditional can be written • If it is false, then a biconditional CAN NOT be written.

  14. Bicondtional: • An angle is acute if and only if it has a measure between 0° and 90°.

  15. Write the conditional as a biconditional statement. • If an animal is a leopard, then it has spots. Write the converse. • If an animal has spots, then is a leopard. False

  16. Therefore a biconditional for this statement does not exist!

  17. More Examples: Write each conditional as a biconditional statement, if possible. Be sure to give a counterexample if the converse is false! Try It!

  18. If is perpendicular to , then their intersection forms a right angle. Converse: If, and intersect at a right angle, then they are perpendicular to each other. True

  19. Biconditional: • is perpendicular to iff their intersection forms a right angle.

  20. Counterexample: • let then 2. If x2 < 49, then x < 7 If x < 7, then x2 < 49. Converse: Therefore, a biconditional can not be written!

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