Geometry Week 1 Notes PDF

Geometry Week 1 – Reasoning and Proofs What comes next in the pattern? How many points are in the bottom row of figure 5? How many points would there be all together in figure 6?  Inductive Reasoning A conjecture is an unproven statement that is based on an observation. You u...

Geometry Week 1 – Reasoning and Proofs What comes next in the pattern? How many points are in the bottom row of figure 5? How many points would there be all together in figure 6?  Inductive Reasoning A conjecture is an unproven statement that is based on an observation. You use inductive reasoning when you find a pattern in specific cases and make a conjecture. Given five collinear points, make a conjecture about the number of ways to connect different pairs of the points:  To show a conjecture is true, you must show it is true for all cases. To show that a conjecture is false, you need only one case where the conjecture is false. This is called a counterexample, or a specific case where the conjecture is false. Example Conjecture: The sum of two numbers is always greater than the larger numbers. True or false? Counterexample: -2 + -3 = -5 and -5 is not greater than -2 Example Find a counterexample to show that the following conjecture is false: Conjecture: The value of x2 is always greater than the value of x. Counterexample:  A conditional statement is a logical statement that has 2 parts, a hypothesis (if) and a conclusion (then). If it is raining, then there are clouds in the sky. Hypothesis Conclusion Re-write the following in if-then form. If x = -3, then 2x + 7 = 1 If a tourist is at the Alamo, then the tourist is in Texas  To prove a conditional statement to be true, you must prove that the conclusion is true every time that the hypothesis is true. It is often easier to show that the conditional statement is false, by finding a one counterexample. Other Types of Statements!  “Negations are the opposite of the original statement” Statement: Today is Wednesday. Negation: Today is not Wednesday. Statement: The cat is not black. Negation: The cat is black. Statement: If something is a bird, then it has feathers. Negation: If something is bird, then it does not have feathers. “Converses” are exchanging the hypothesis and conclusion of the conditional statement Statement: If something is a bird, then it has feathers. Converse: If something has feathers, then it is a bird. Statement: If a dog is a great Dane, then it is large. Converse: If a dog is large, then it is a Great Dane. Statement: If an angle measures 90 degrees, then it is a right angle. Converse: If an angle is a right angle, then the angle measures 90 degrees  “Inverses” are negating both the hypothesis and conclusion of the conditional statement Statement: If something is a bird, then it has feathers. Inverse: If something is not a bird, then it does not have feathers. Statement: If a dog is a Great Dane, then it is large. Inverse: If a dog is not a Great Dane, then it is not large. Statement: If an angle measures 90 degrees, then it is a right angle. Inverse: If an angle does not measure 90 degrees, then the angle is not a right angle.  “Contrapositive” – First, write the converse of the conditional statement. Second, negate both the hypothesis and conclusion of the converse. Statement: If something is a bird, then it has feathers. Contrapositive: If something does not have feathers, then it is not a bird. Statement: If a dog is a Great Dane, then it is large. Contrapositive: If a dog is not large, then it is not a Great Dane. Statement: If an angle measures 90 degrees, then it is a right angle. Contrapositive: If an angle is not a right angle, then the angle does not measure 90 degrees. Determine if the Statement is True Conditional statement: If you are in Academic Challenge, then you are a smart student. True Converse: If you are a smart student, then you are in Academic Challenge. False, a student could be smart but not select Academic Challenge Inverse: If you are not in Academic Challenge then you are not a smart student. False, even if a student is smart, the student does not need to take Academic Challenge courses. Contrapositive: If you are not a smart student then you are not in Academic Challenge. True, the student can’t pass the required math test to qualify for Academic Challenge  A conditional statement and its contrapositive are either both true or both false. Likewise , the converse and inverse of a conditional statement are either both true or both false. Pairs of statements like these (both true or both false) are called equivalent statements.  “Soccer Players are Athletes” 1. Construct a conditional statement for the conjecture If you are a soccer player, then you are an athlete. 2. Is the conditional statement true? TRUE 3. State the converse. Is the converse true? If you are an athlete, then you are a soccer player….. FALSE 4. State the inverse. Is the inverse true? If you are not a soccer player, then you are not an athlete…. FALSE 5. State the contrapositive. Is the contrapositive true? If you are not an athlete, then you are not a soccer player…. TRUE If a conditional statement and its converse are true, the two statements can be combined into a biconditional statement. This is done by replacing “then” in the conditional statement by “if and only if”. Example - Conditional statement for perpendicular lines – If two lines intersect to form a right angle, then they are perpendicular lines. (True since this a definition of perpendicular lines) Converse - If two lines are perpendicular, then they intersect to form a right angle. (True, another definition of perpendicular lines) Since both are true we can combine into a biconditional statement about perpendicular line – Two lines intersect to form a right angle if and only if the two lines are perpendicular.  Write Biconditional Statements 1. If Mary is in theater class, she will be in the fall play. If Mary is in the fall play, she must be taking theater class Mary is in theater class if and only if Mary is in the fall play 2. An angle with a measure between 90⁰ and 180⁰ is called obtuse An angle is obtuse if and only if the angle measures between 90⁰ and 180⁰ 3. In equilateral polygons all sides are congruent A polygon is equilateral if and only if all of the sides are congruent Break!!!!! Then work on this ☺  Deductive Reasoning uses facts, definitions, accepted geometrical and mathematical properties, and the laws of logic to form a logical argument or proof. Proofs in Geometry are based on deductive reasoning.  Logical Law of Detachment Law of detachment – If the hypothesis of a true conditional statement is true, then the conclusion is also true. Everyday example: Mary goes to the movies every Friday and Saturday night. Today is Friday. Hypothesis – If it is Friday or Saturday night, then Mary goes to the movies. Today is Friday SO you can conclude that Mary will go to the movies. Be careful --- When the hypothesis is not true, you can not draw a logical conclusion  Angle fits the hypothesis, so you can conclude that the angle is obtuse nothing, the angle is not between 90 and 180 degrees, so the angle does not fit or make the hypothesis true. Can‘t use the law of detachment to conclude anything.  Logic Law of Syllogism Law of Syllogism --- enables you to combine true conditional statements into new conditional statement: – If hypothesis p, then conclusion q. If these are true, – If hypothesis q, then conclusion r. – If hypothesis p, then conclusion r. Then this is true. Example: 1. If you play a clarinet, you play a woodwind instrument 2. If you play a woodwind instrument, you are a musician “If you play a clarinet, then you are a musician” Practice the Laws of Logic Angle A is a right angle! Rick will get an A in Chemistry! Angle R is obtuse!! x2 > 20 Intro to Proofs In Geometry, we spend a lot of time PROVING certain statements to be true. There are a few ways this can be done, but the process is not unlike JUSTIFYING your steps when you solve an equation. We use already proven statements or properties as our justification for each step in the process!  These are some of the properties that you’ve used throughout Algebra 1. Addition Property => if a = b, then a + c = b+ c 2. Subtraction Property => if a = b, then a – c = b - c 3. Multiplication Property => if a = b, then ac = bc 4. Division Property => if a = b, then a/c = b/c if c ≠ 0 5. Substitution Property => if a = b, then b can be substituted for a in equations or expressions 6. Distributive Property => a(b+c) = ab + ac  The most conventional type of proof is the 2-Column Proof 2-Column Proof, where you list your process in the left column and your justification in the right. Solve 3(2x + 11) = 9 Steps to Solve Property to Justify 3(2xDivision + 11) =9 Property Given fact Division Property 6x +Division 33 =Property 9 Distribution Property Division Property 6x Division = -24Property Subtraction Property Division Property x =Division -4 Property Division DivisionProperty Property  Solve 2x + 5 = 20 – 3x Steps to Solve Property to Justify 2x + Division 5 = 20 – 3x Property Given fact Division Property 2x + 5 +Division 3x =Property 20 – 3x + 3x AdditionDivision PropProperty of Equality 5x + Division 5 = 20 Property SimplifyDivision Property 5x =Division 15 Property Subtraction Prop of Equ Division Property x=3 Division Property DivisionDivision PropProperty of Equality Other properties of real numbers that apply to Geometry: Reflexive Property Symmetric Property Just like the Law of Syllogism. Transitive Property If a = b and b = c, then a = c  Given: g = 2h, g + h = k, k = m Prove: m = 3h Steps to Solve Property to Justify g = 2h Division Property Given g + h = kDivision Property Given k = mDivision Property Given 2h + Division h = kProperty Substitution Division Property 3h = k Division Property CombineDivision LikeProperty Terms 3h = m Division Property Transitive Property Division Property  Given: m = n + 5, 2m = n Prove: m = -5 Steps to Solve Property to Justify m=n + 5Property Division Given 2m = n Division Property Given m = 2m +Property Division 5 Substitution Division Property -m = 5 Division Property Subtraction Prop of Eq Division Property m = -5 Division Property Division Division PropProperty of Eq  Postulates In Geometry, in addition to Properties, we have Postulates – which are statements that we know to be true: Angle Addition Postulate: The sum of two angle measures that are joined by a common ray will be equal to the measure of the angle they form together. Segment Addition Postulate: The sum of the distance from a point on a segment to each endpoint is equal to the length of the segment itself.  Basic Geometric Proofs Statements Property to Justification Division Property Given fact Division Property Division Property Angle Addition Postulate Division Property Division Property Substitution Division Property Division Property Angle Addition Postulate Division Property Division Property Transitive Property Division Property  Basic Geometric Proofs Given: AB = CD Prove: AC = BD Statements Property to Justify AB =Division CD Property Given fact Division Property AC = ABDivision + BCProperty SegmentDivision Addition Property Post AC = CD + BC Division Property Substitution Division Property CD + BC = BD Division Property SegmentDivision Addition Property Post AC = BDDivision Property Transitive Property Division Property  Challenge Statements Property to Justify Given Division Property Division Property Given Division Property Division Property Division Property Division Angle Addition Property Postulate Angle Addition Postulate Division Property Division Property Division Property SubstitutionDivision Property Subtraction Prop of Eq