1. GEOMETRIC PROOFS A Keystone Geometry Mini-Unit

2. Geometric Proofs – An Intro Why do we have to learn “Proofs”? A proof is an argument, a justification, a reason that something is true. It’s got to be a particular kind of reasoning – logical – to be called a proof. There are certainly plenty of other, equally valid forms of reasoning like inductive reasoning or deductive reasoning. Some of them are even used in “doing” mathematics. But they’re not proofs. A proof is just the answer to the question “Why?” 2

3. Conditional Statements If-then statements are also called conditional statements or simply conditionals. An if-then statement has two parts, the hypothesis and the conclusion. Example: If it rains after school, then I will give you a ride home. Example: If B is between A and C, then AB + BC = AC. If p, then q. 3

4. Practice State the hypothesis and the conclusion of each conditional. 4

5. Different Forms of a Conditional The following all have the same hypothesis and conclusion. If p, then q. p implies q. p only if q. q if p. Examples: If it rains after school, then I will give you a ride home. It rains after school implies I will give you a ride home. It rains after school only if I will give you a ride home. I will give you a ride home if it rains after school. 5

6. Converse, Inverse and Contrapositive of a Conditional 6 Conditional: If p, then q. If it rains after school, then I will give you a ride home. Converse: If q, then p. If I will give you a ride home, then it rains after school. Inverse: If not p, then not q. If it will not rain after school, then I will not give you a ride home. Contrapositive: If not q, then not p. If I will not give you a ride home, then it will not rain after school.

7. Example: Identify the hypothesis: Identify the conclusion: Is your conditional statement true or false? Write the converse: Is your converse statement true or false?, 7

8. 8 Converse: If Ed lives south of Canada, then Ed lives in Texas. Inverse: If Ed does not live in Texas, then Ed does not live south of Canada. Contrapositive: If Ed does not live south of Canada, then Ed does not live in Texas. Conditional: If Ed lives in Texas, then Ed lives south of Canada. Write the converse, inverse and contrapositive given the following conditional. State if each statement is true or false.

9. Counterexample Counterexample: Is an example that gives a true hypothesis but a false conclusion. If you state something that is false, you cannot just say so, you must PROVE that it is false by giving a counterexample. A counterexample can be in the form of a verbal statement, a mathematical problem, or as a visual picture. 9

10. Example: This conditional statement was TRUE. But the converse, was FALSE. PROVE IT: Converse: 10

11. 11 Converse: If Ed lives south of Canada, then Ed lives in Texas.FALSE, he lives in PA. Inverse: If Ed does not live in Texas, then Ed does not live south of Canada.FALSE, he lives in PA. Contrapositive: If Ed does not live south of Canada, then Ed does not live in Texas. TRUE Conditional: If Ed lives in Texas, then Ed lives south of Canada.TRUE Provide counterexamples for the false statements.

12. What does “Logically Equivalent” mean? Logically Equivalent, is when two statements have the same outcome. A conditional and its contrapositive are logically equivalent. If the conditional is true, then the contrapositive is also true. The converse and inverse of a conditional are logically equivalent. If the converse is true, then the inverse is also true. 12

13. What about a Biconditional? If a conditional and its converse are true they can be combined into one single statement using the words “if and only if”, which is called a biconditional. Biconditional: p if and only if q. Most of our definitions are biconditional statements. Example: Definition of Congruent Segments: Congruent segments are segments that have equal lengths. Biconditional: Segments are congruent if and only if their lengths are equal. 13