GEOMETRIC PROOFS A Keystone Geometry Mini-Unit. Geometric Proofs – An Intro Why do we have to learn “Proofs”? A proof is an argument, a justification,

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Presentation transcript:

GEOMETRIC PROOFS A Keystone Geometry Mini-Unit

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

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

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

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

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.

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 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.

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

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

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.

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

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