Mr. Schaabs Geometry Class Our Lady of Providence Jr.-Sr. High School Deductive reasoning uses facts, definitions, accepted properties and the laws of logic to form a logical argument. Remember: Inductive reasoning uses specific examples and patterns to form the same kind of logical argument; knowing the difference between the two is very important! Property Characteristic or quality that something has. Example: Addition Property - you can add any number to both sides of an equation. Definition The reason behind the creation of a term Example: Definition of midpoint point that splits a segment into two congruent segments. Theorem A rule that has been proven to be true using known properties. Example: Triangle Sum Theorem: the interior angles of a triangle always add up to 180. PropertyDescriptionExample Addition PropertyYou can add a value to both sides of an equation. Subtraction PropertyYou can subtract a value from both sides of an equation. Multiplication PropertyYou can multiply both sides of an equation by a value. Division PropertyYou can divide both sides of an equation by a value. PropertyDescriptionExample Substitution PropertyYou can swap out one value for an equivalent value. Given: y = 2x 6 and x = 5 You can deduce: Reflexive PropertyA value is always equal to itself. (Duh!) Given: 30 You can deduce: Symmetric PropertyYou can switch the sides of an equation. Given: 7 = x You can deduce: Transitive PropertyIf a = b and b = c, then a = c (also known as the chain rule) Given: RS TV and TV LM You can deduce: PropertyDescriptionExample Distributive PropertyYou can distribute a factor to every term in parentheses. Given: 12 = 2(3x - 4) You can deduce: Combining Like TermsYou can add or subtract two terms if their variables are identical. Given: 18 = 3x 9x You can deduce: SimplificationYou can perform mathematical operations to an expression. Given: 2 = 3(4) 10 You can deduce: Parts of a two-column proof: Given line Informs you of all of the known facts Always the first line of your proof! Prove line Informs you of what information you are being asked to prove Always the last line of your proof! Statements Column 1; always on left Lists every logical step in your argument Reasons Column 2; always on the right The justification for why you deduced each statement; informs the reader why each statement is true. Given Subtraction Property Distributive Property Addition Property Division Property Algebraic Proof: Proof that involves solving an equation and justifying each step with the Properties of Equality. Geometric Proof: Proof that involves using theorems and definitions to prove a statement about geometric figures. Typically includes a diagram. Given Subtraction Prop. Division Prop. Subtraction Prop. Mult. Prop. Or Div. Prop 4x 2(2 - x) = 4x 24 -2(2 x) = x = 12 -x = 10 x = -10 3x + 4y = 23 x = 1 Given Substitution Property Simplify Subtraction Property Division Property 3(1) + 4y = y = 23 4y = 20 y = 5 HW 2-3: Complete Practice Problems #1- 11 ALL (entire first page) and App. #1-6 ALL Homework Quiz next class over algebraic Proofs! To write a geometric proof, you need to have strong knowledge all of the rules, theorems, and definitions that apply to each type of figure! Example: What do you know about vertical angles? They are always congruent What do you know about an angle bisector? It splits an angle into two congruent angles What does the segment addition postulate tell us? If a segment is made up of two or more smaller segments, you can add the parts together to equal the whole. The Segment Addition Postulate: If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC The Angle Addition Postulate: If point B is in the interior of AOC, then m AOB + m BOC = m AOC Given Def. of Vertical Angles Substitution Property 1 4 2 4 1 3 2 1 2 3 Def. of Vertical Angles Given Def. of Vertical Angles Substitution Property 2 5 2 4 5 8 4 5 4 8 Def. of Vertical Angles Given Def. of Supplementary Angles Substitution Property 1 and 2 are supplementary Def. of Supplementary Angles Def. of Congruent Angles 2 and 3 are supplementary m 1 + m 2 = 180 m 2 + m 3 = 180 m 1 + m 2 = m 2 + m 3 m 1 = m 3 1 3 Subtraction Property Given Def. of Vertical Angles Symmetric Property Substitution Property 4 6 6 4 1 4 3 6 3 4 1 3 Substitution Property Youll need: Piece of paper Pen or pencil Writing Prompt: think back to a memorable argument you had with someone else in your life. Take a few moments to recreate the argument in your head and try to remember everything that was said. Address the following questions: Was there a clear winner of the argument? If so, did you win or lose? What good points did the winner bring up to help them convince the other side that they were right? What points could the person on the losing end have made to help convince the other side? If there was not a clear winner, why not? Do you think you could have won the argument if you had done or said something different? Now organize your thoughts about the points you made in the argument as well as your opponents points. Write a paragraph proof that sums up all of your ideas on why you should be right. Paragraph Proof: When I was in high school, my older sister and I had the classic household argument. She insisted that my brother and I should put the toilet seat down after we were done using the restroom and I argued that it is not the males responsibility to lower the seat, but that both men and women are equally responsible for making sure the seat is in the proper position before using the restroom. Who do you think is right? Why? MY ARGUMENT:MY SISTERS ARGUMENT: There are three men in the house and only two women, so on average a man is more likely to use the toilet next. Male house members should be commended for putting the seat up, avoiding sprinkles on the seat and creating a more sanitary environment. I view women as equal to men in every respect, and therefore giving women special treatment is a form of discrimination, much like offering to carry something heavy for a woman. Leaving the seat up helps to form good preparation habits, so that both men and women are more observant when a toilet seat is in the incorrect position. Men are the ones who choose to lift the seat rather than sitting down, so they should leave it as the found it. A person could fall into the toilet, potentially hurting themselves, so the seat should always be in the down position. The rim of the toilet is unsightly and should not be displayed in the event that a guest uses the restroom. Men also need to sit sometimes, so it benefits both parties to an extent. Given: In my household, men tend to leave the toilet seat up, women tend to leave it down. Prove: It should be left up. Men tend to leave the toilet seat up when they are finished using the restroom, and women seem to prefer that it be left down. However, by leaving the seat up, men avoid sprinkling the seat and thus creating an unsanitary environment. Lowering the seat upon completion does not make mathematical sense in a household where men outnumber women, because more often than not it will need to put right back up. In order to foster a sense of autonomy and attentiveness, It should be each persons responsibility to prepare the toilet according to their own needs each time it is used. If your work is done for you, you are more likely to become lazy and assumptive, which will eventually lead to an embarrassing or potentially harmful incident. Therefore, the toilet seat should be left in its position of last use and neither men or women should be held responsible for the others toilet safety.