Biconditionals & Deductive Reasoning Geometry

Biconditionals When a conditional and its converse are true, you can combine them as a true biconditional. You do this by connecting the two parts of the conditional with the words “if and only if” If two angles have the same measure, then the angles are congruent. If two angles are congruent, then the angles have the same measure. Since both of these are true, the biconditional statement would read: Two angles are congruent if and only if the angles have the same measure.

Biconditionals Write the converse of this statement. If both statements are true, then rewrite it as a biconditional. In the United States, if it is July 4th, then it is Independence Day.

Separating a Biconditional into Parts You can also rewrite a biconditional as a conditional and its converse. A number is divisible by 3 if and only if the sum of its digits is divisible by 3. This would be separated into the two statements: If a number is divisible by 3, then the sum of its digits is divisible by 3. If the sum of a number’s digits is divisible by 3, then it is divisible by 3 .

Separating a Biconditional into Parts Write the 2 statements that form the biconditional: A line bisects a segment if and only if the line intersects the segment only at its midpoint.

Definitions A good definition has several components: It uses clearly understood terms. It is precise and specific. (avoid opinion and use of words such as large, sort of, and almost). It is reversible. This means that a good definition can be written as a true biconditional. Definition of Perpendicular Lines: Perpendicular Lines are two lines that intersect to form right angles. Conditional: If two lines are perpendicular, then they intersect to form right angles. Converse: If two lines intersect to form right angles, then they are perpendicular. Since the two are both true, the definition can be written as a true biconditional: Two lines are perpendicular if and only if they intersect to form right angles.

Definitions A dog is a good pet. Parallel lines do not intersect. An angle bisector is a ray that divides an angle into two congruent angles.

Chapter 2.4 Deductive Reasoning

Vocabulary Deductive Reasoning (logical reasoning) = the process of reasoning logically from given statements to a conclusion. You conclude from facts.

Example A physician diagnosing a patient’s illness uses deductive reasoning. They gather the facts before concluding what illness it is.

Law of Detachment Law of Detachment = If a conditional is true and its hypothesis is true, then its conclusion is true In symbolic form: If p q is a true statement and p is true, then q is true

Law of Detachment Example Use Law of Detachment to draw a conclusion If a student gets an A on a final exam, then the student will pass the course. Felicia gets an A on the music theory final exam Conclusion: She will pass Music Theory class

Law of Detachment: Try it Use Law of Detachment to draw a conclusion If it’s snowing, then the temperature is below 0 degrees It is snowing outside Conclusion: It’s below 0 degrees

Law of Detachment Example: Careful! Use the Law of Detachment to draw a conclusion If a road is icy, then driving conditions are hazardous Driving conditions are hazardous Conclusion: You can’t conclude anything!!!

Quick Summary You can only conclude if you are given the hypothesis!!! If you are given the conclusion you CAN NOT prove the hypothesis

Law of Detachment What conclusion can you draw from the following two statements? If a person doesn’t get enough sleep, they will be tired. Evan doesn’t get enough sleep. Evan will get more sleep. Evan will be tired Evan will not be tired. You cannot make a conclusion.

Law of Syllogism If pq and qr are true statements, then pr is a true statement Example: If the electric power is cut, then the refrigerator does not work. If the refrigerator does not work, then the food is spoiled. So if the electric power is cut, then the food is spoiled.  

Law of Syllogism example: Use law of syllogism to draw a conclusion If you are studying botany, then you are studying biology. If you are studying biology, then you are studying a science Conclusion: If you are studying botany, then you are studying a science

Law of Syllogism: Try it Use the law of Syllogism and draw a conclusion If you play video games, then you don’t do your homework If you don’t do your homework, then your grade will drop Conclusion: If you play video games, then your grade will

Law of Syllogism What conclusion can you draw from the following two statements? If you have a job, then you have an income. If you have an income, then you must pay taxes. If you have a job then you must pay taxes If you don’t have a job, then you don’t pay taxes. If you pay taxes, then you have a job. If you have a job, then you don’t pay taxes.