2-3 Biconditionals and Definitions

Quick Review

When a conditional and its converse are true, you can combine them as a true biconditional using the phrase if and only if. The symbolic form of a biconditional is p ↔ q . p left right arrow q .  You can write a good definition as a true biconditional.

Example

Is the following definition reversible? If yes, write it as a true biconditional.

A hexagon is a polygon with exactly six sides.

Yes. The conditional is true: If a figure is a hexagon, then it is a polygon with exactly six sides. Its converse is also true: If a figure is a polygon with exactly six sides, then it is a hexagon.

Biconditional: A figure is a hexagon if and only if it is a polygon with exactly six sides.

Exercises

Determine whether each statement is a good definition. If not, explain.

22. A newspaper has articles you read.
23. A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays.
24. An angle is a geometric figure.
25. Write the following definition as a biconditional. An oxymoron is a phrase that contains contradictory terms.

26. Write the following biconditional as two statements, a conditional and its converse.

    Two angles are complementary if and only if the sum of their measures is 90.

2-4 Deductive Reasoning

Quick Review

Deductive reasoning is the process of reasoning logically from given statements to a conclusion.

Law of Detachment: If p → q p rightwards arrow q  is true and p is true, then q is true.

Law of Syllogism: If p → q p rightwards arrow q  and q → r q rightwards arrow r  are true, then p → r p rightwards arrow r  is true.

Example

What can you conclude from the given information? Given: If you play hockey, then you are on the team. If you are on the team, then you are a varsity athlete.

The conclusion of the first statement matches the hypothesis of the second statement. Use the Law of Syllogism to conclude: If you play hockey, then you are a varsity athlete.

Exercises

Use the Law of Detachment to make a conclusion.

27. If you practice tennis every day, then you will become a better player. Colin practices tennis every day.
28. ∠ 1 angle 1  and ∠ 2 angle 2  are supplementary. If two angles are supplementary, then the sum of their measures is 180.

Use the Law of Syllogism to make a conclusion.

29. If two angles are vertical, then they are congruent. If two angles are congruent, then their measures are equal.
30. If your father buys new gardening gloves, then he will work in his garden. If he works in his garden, then he will plant tomatoes.

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