Formal Geometry. Conditional Statements
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1 Formal Geometry Conditional Statements
2 Objectives Can you analyze statements in if then form? Can you write the converse, inverse, and contrapositive of if then statements?
3 Inductive Reasoning Inductive reasoning is reasoning that uses a number of examples to arrive at a conclusion (example: a pattern) Conjecture: a hypothesis Counterexample: a false example; an example that shows a conjecture is untrue
4 Conditional Statement A conditional statement is a statement written in if-then format. Symbolic notation: p q and is read as if p then q Example: If it stops raining then you can go outside. Hypothesis of a conditional statement is the if part; called p Conclusion of a conditional statement is the then part; called q
5 Related Conditionals Example: If it stops raining then you can go outside. Converse: exchange the hypothesis and conclusion; if q then p or q p Inverse: negate both the hypothesis and conclusion if not p then not q or ~p ~q ~ means not Contrapositive: negate both the hypothesis and conclusion of the converse if not q then not p or ~q ~p
6 Biconditional Statements A statement that can be used if the converse and conditional are both true Uses if and only if (iff) All definitions are biconditional Example: An angle is a right angle if and only if it measures 90 degrees.
7 Logically Equivalent Statements are logically equivalent if they have the same truth value (they say the same thing) Conditional and Contrapositive are logically equivalent Converse and inverse are logically equivalent
8 Logically Equivalent Example Write the inverse, converse, and contrapositive of the statement and determine the truth value. If a shape is a square then it is a rectangle. Statement: if a shape is a square, then it is a rectangle (T) Converse: If a shape is rectangle, then it is square (F) Inverse: If a shape is not a square, then it is not a rectangle (F) Contrapositive: If a shape is not a rectangle, then it is not a square (T) When the converse is true, the inverse will be true. When the converse is false, the inverse will be false. (logically equivalent!)
9 Example Make a conjecture about each pattern or geometric relationship. 1) -2, 0, 2, 4, 6, 8 2) Point P is the midpoint of segment NQ.
10 Examples Write each statement in if-then form. A mammal is a warm-blooded animal. The sum of the measures of two supplementary angles is 180 degrees.
11 Examples Identify the hypothesis and conclusion of the following statement. If today is Wednesday, then tomorrow is Thursday.
12 Examples Determine the truth value of the statement. If false find a counter example. If an animal is spotted then it is a Dalmatian. If angle A and angle B are complementary angles, then they share a common side.
13 Example Write the converse, inverse, and contrapositive of the statement. State whether the statement is True or False. If false give a counterexample. If an animal is a lion then it can roar.
14 Warm Up 1) Identify the hypothesis and conclusion of the following statement. If today is Wednesday, then tomorrow is Thursday. 2) Write the statement in if-then form. A mammal is a warm-blooded animal. 3) Determine the truth value of the statement. If false find a counter example. If an animal is spotted then it is a Dalmatian. If angle A and angle B are complementary angles, then they share a common side. 4) Write the converse, inverse, and contrapositive of the statement. State whether the statement is True or False. If false give a counterexample. If an animal is a lion then it can roar.
15 Biconditional Statements A statement that can be used if the converse and conditional are both true Uses if and only if (iff) All definitions are biconditional Example: An angle is a right angle if and only if it measures 90 degrees.
16 Example Rewrite each pair of statements in biconditional form. If lines are coplanar and do not intersect, then they are parallel lines. If lines are parallel, then they are coplanar lines and do not intersect. Biconditional:
17 Example Rewrite the biconditional into two statements. Two angles are complementary if and only if their measures sum to 90.
18 Logical Conclusions A valid conclusion can be made if the given follows the conditional or contrapositive statement!
19 Example Write the conclusion to the statement given. If Kelly was swimming, then she was swimming in a pool. Kelly was swimming. Conclusion:
20 Example If the statements follow logically valid reasoning patterns, write the conclusion. If the statements do not fit valid reasoning patterns, write no valid conclusion. In either case, identify the conditional statement being used (i.e., converse, inverse, contrapositive). If a bird is an ostrich, then it cannot fly. The bird is an ostrich. Conclusion: Conditional: If it rains outside, I will wear my coat. It is not raining. Conclusion: Conditional: Answer: bird cannot fly; conditional statement Answer: no valid conclusion; inverse
21 Examples If Mika goes to the beach she will wear sunscreen. Mika wears sunscreen Conclusion: Conditional: If a student turns in a permission slip, then the student can go on the field trip. The student cannot go on the field trip. Conclusion: Conditional: Answer: no valid conclusion; converse Answer: student did not turn in permission slip; contrapositive
22 Example If I went to the movies, then it is Friday. If it is Friday, then I got paid. Conclusion:
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