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1 Geometry Definitions, Postulates, and Theorems hapter 2: Reasoning and Proof Section 2.1: Use Inductive Reasoning Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. onjecture - n unproven statement that is based on observations and patterns. Ex. What s next? Visual Pattern a) b) c) Number Pattern a) 2, 5, 8, 11, b) 5, 7, 11, 17, 25, Inductive Reasoning - Process of looking for patterns and making conjectures. Ex. The product of an odd number and an even number is. 14 add 3 to preceding number 35 add 2, then 4, then 6,... even What is the pattern observed? 3(8) = 24 6(5) = 30 11(24) 264 7(10)= 70 Ex. The sum of an odd number and an even number is odd = = = = 31 ounterexample n example that shows that a conjecture is false. Ex. The difference between two positive numbers is always positive. Find a counterexample. (there are many correct answers) 3-9 = -6

2 Section 2.2: nalyze onditional Statements Standards: 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. onditional Statement logical statement that has two parts, a hypothesis and a conclusion. conditional statement is best when written in IF..., THEN... form. Ex. Rewrite the conditional statements in if-then form: Hypothesis The "IF" part of a conditional statement. onclusion The "THEN" part of a conditional statement. If it is 6pm, then it is time for dinner. a) It is time for dinner if it is 6pm. If an animal is a monkey, then it has a tail. b) ll monkeys have tails. *onditional statements can be true or false. Decide whether the statement is true or false. If false, provide a counterexample. Ex. If a number is odd, then it is divisible by 3. onverse statement formed by switching the hypothesis and conclusion of a conditional statement. NOTE: converse is not always a true statement. Ex. Statement: If you see lightning, then you hear thunder. f you hear thunder, then you see lightening. onverse: Negation Writing the opposite or negative of the statement. Ex. a) m = 35 D is not obtuse m 35, b) D is obtuse, Inverse The opposite of the hypothesis ND the opposite of the conclusion of a conditional statement. Ex. Statement: If an animal is a dog, then it has four legs. If an animal is not a dog, then it does not have four legs. Inverse: ontrapositive The opposite of the hypothesis ND the opposite of the conclusion of a converse. Ex. Statement: If an animal is a fish, then it can swim. ontrapositive: If an animal cannot swim, then it is not a fish. (over)

3 Ex. a) Statement: If Equivalent Statements - Statements that are both true or both false. o m 30, then is acute. both are true ontrapositive: If is not acute, then m = 30. If m = 30, then is not acute. b) Inverse: both are false onverse: If is acute, then m = 30 ***Perpendicular Lines IF two lines intersect to form a right angle, THEN the lines are perpendicular. D notation means perpendicular D iconditional Statement statement that contains the phrase "if and only if." This is equivalent to a conditional statement and it's converse. iconditional Statement: x + 3 = 5 if and only if x = 2. If x + 3 = 5, then x = 2. *an be written as "iff" onditional: T/F onverse: T/F If x = 2, then x + 3 = 5. The biconditional statement is true since the conditional statement is true and the converse is true. iconditional Statement: x 2 = 4x if and only if x = 4. 2 If x = 4x, then x = 4. onditional: T/F 2 onverse: T/F If x = 4, then x = 4x. The biconditional statement is false since the conditional statement is falseand the converse is.

4 Section 2.3: pply Deductive Reasoning Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Deductive Reasoning relies on facts, definitions and accepted properties in logical order to write a logical argument. Inductive Reasoning examples and patterns are used to form a conjecture. *Two types of Deductive Reasoning: Law of Detachment IF p q is true, ND p is true, THEN q is true. Ex. If m < 90, then is acute. that m DEF = 42, we can conlude that DEF is acute. Law of Syllogism IF p q is true, ND q r is true, THEN p r is true. *Like the transitive property (cut out the middle man) Ex. If m = 45, then m < 90. If m <90, then is acute. If m = 45, then is acute. Ex. Decide whether deductive or inductive reasoning is being used in the following argument. Everyday Mrs. Maya reads off the homework answers, then goes over the wrong ones on the board. Today she has read off the homework answers, so you can conclude that she will go over the wrong answers on the board. Inductive Why did you decide so? ased on patterns. Ex. Law of Syllogism p q If you are a student, then you have lots of homework. q r If you have lots of homework, then you have little social life. p r If you are a student, then you have little social life.. (over)

5 Section 2.3 Extension Symbolic Notation: p represents the hypotheses q represents the conclusion conditional statement (if-then): p q converse q p negation of p ~ p inverse ~ p ~ q contrapositive ~ q ~ p biconditional p q (means p q and q p ) Ex. Use p and q to write the symbolic statement in words. p: number is divisible by 3. q: number is divisible by 6. p q If a number is divisible by 3, then it is divisible by 6. q p If a number is divisible by 6, then it is divisible by 3. ~ p ~ p ~ q number is not divisible by 3. If a number is not divisible by 3, then it is not divisible by 6. ~ q ~ p If a number is not divisible by 6, then it is not divisible by 3. p q number is divisible by 3 if and only if a number is divisible by 6.

6 Section 2.4: Use Postulates and Diagrams Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. Postulate 5 IF you are given any two points, THEN there exists exactly one line. Postulate 6 IF you are given a line, THEN it contains at least two points. U and are on line Line contains & Postulate 7 IF two lines intersect, THEN their intersection is exactly one point. Lines and intersect at point P. P Postulate 8 IF you are given any three noncollinear points, THEN there exists exactly one plane. R Q Postulate 9 IF you are given a plane, THEN it contains at least three noncollinear points. Postulate 10 IF two points lie in a plane, THEN the line containing them also lies in the plane. P, Q, and R are on plane. (or plane PRQ) Plane contains points,, and c. and are on plane, then lies on plane. M R Postulate 11 IF two planes intersect, THEN the intersection is a line. R M Planes M and R intersect at line. Line Perpendicular to a Plane line that intersects a plane at a point and makes a right angle with the plane. Ex. State the postulate that verifies the truth of the statement: V and T lie on line n. Write out the postulate. n V S T T ST plane *a plane has infinitely many lines Postulate 5: Through any two points, there is exactly one line.

7 Section 2.5: Reason Using Properties from lgebra Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. ddition Property of Equality IF a = b, THEN a + c = b + c. x - 2 = 10 x = 12 Subtraction Property of Equality IF a = b, THEN a c = b c. 5 + n = 8 n = 3 Multiplication Property of Equality IF a = b, THEN ac = bc. x/2 = 4 x = 8 Division Property of Equality IF a = b, THEN a c = b c. 4y = 20 y = 5 ***Substitution Property of Equality IF a = b, THEN a can be substituted for b anytime. If m = 100, and m = m, then m = 100. Distributive Property 2(x + 4) = 2x + 8 or 5x - 3x = x(5x - 3) add 2 to both sides subtract 5 from both sides multiply both sides by 2 Divide both sides by 4 a ( b c) ab ac OR ab ac a ( b c) a ( b c) ab ac OR ab ac a ( b c) Properties of Equality for Real Numbers, Segments and for ngles Real Numbers Segments ngles ***Reflexive IF a is a real # IF Segment, IF ngle, THEN a = a THEN =. THEN m m. ***Symmetric IF a = b IF D, IF m m, THEN b = a THEN D. THEN m m. ***Transitive IF a = b IF D, IF m m, ND b = c, ND D E F, ND m m, THEN a = c. THEN EF. THEN m m. *cut out the middle man 10 = 10 x = x If 10 = x, then x = 10. If a = x + 3, and x + 3 = 7, then a = 7. (over)

8 Ex. lgebraic Proof Solve the equation 3x 12 8x 18 and state the reason for each step. Statements 3x + 12 = 8x = 5x = 5x = x 4. x = Reasons Subtraction Prop. of = ddition Prop. of = Division Prop. of = Symmetric Prop. Ex. Geometric Proof Show the perimeter of is equal to the perimeter of D. D Statements Reasons D = ; D = = = Perimeter of Definition of Perimeter D + D + = Perimeter of D 4. D + D + = Perimeter of 4. Substitution 5. Perimetr of D = Perimetr of 5. Reflexive Substitution Ex. Use the property to complete the statement. Transitive Property: If YZ D and D JK, then YZ = JK

9 Section 2.6 Prove Statements about Segments & ngles Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 2.0 Students write geometric proofs, including proofs by contradiction. Proof logical argument that shows a statement is true. Two-olumn Proof Numbered statements and corresponding reasons that show an argument in a logical order. Statements Reasons 1) 1) 2) 2) etc) etc) Theorem statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs. ***Theorem 2.1 ongruence of Segments is Reflexive GIVEN: Segment, THEN. ***Theorem 2.1 ongruence of Segments is Symmetric IF D, THEN D. ***Theorem 2.1 ongruence of Segments is Transitive IF D, ND D EF, THEN EF. Ex. : PQ XY P Q Prove: XY PQ X Y Statement Reason PQ = XY PQ = XY 2. XY = PQ Def. of = Segments Symmetric Property (over)

10 Ex. : Q is the midpoint of PR. P Q R 1 Prove: PQ = PR 2 1. Q is the midpoint of PQ Definition of Midpoint 3.PR = PQ +QR PR = PQ +PQ, PR = 2PQ 4. Substitution 5. PQ = QR 1 2 PR = PQ 5. PQ = PR *ombine like terms Seg. dd. Post. Division Prop. of = Symmetric Prop. Ex. : E F = GH Prove: EG = F H E F G H EF = GH Reflexive 2. FG = FG EF+FG=GH+FG 3. dd. Prop. of = 4. EG = EF + FG, FH = GH + FG 4. Segment ddition Postulate 5. EG = FH 5. Substitution Ex. : H H, H 12 Prove: H H 1. H H, H H = H Definition of ongruent Segments = H 3. Substitution 4. H = Symmetric Prop. D Ex. In the diagram, if and D, find. 3x - 1 = 2x + 3 x - 1 = 3 x = 4 = 2x + 3 = 2( 4 ) + 3 = 11 3x 1 2x 3 D

11 ***Theorem 2.2 ongruence of ngles is Reflexive GIVEN: ngle, THEN. (over) ***Theorem 2.2 ongruence of ngles is Symmetric IF, THEN. ***Theorem 2.2 ongruence of ngles is Transitive IF, ND, THEN. Ex. : Prove: Proof of Transitive Property of ongruence, Statements Reasons 1), 1) given 2) m m, m m 2) 3) m m 3) 4) 4) Def. of = angles Transitive Property Def. of = angles Ex. : 1 2, 3 4, 2 3 Prove: , 3 4, = 4 Transitive Transitive Ex. : m 1 63, 1 3, 3 4 Prove: m m 1 63, 1 3, = 4 2. Transitive Property 3. m 1 = m 4 3. Def. of = angles = m 4 4. Substitution 5. m 4 = Symmetric

12 Section 2.7 Prove ngle Pair Relationships Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 2.0 Students write geometric proofs, including proofs by contradiction. ***Theorem 2.3 Right ngle ongruence Theorem ll right angles are congruent. Ex. : D and are right angles, D Prove: D D D 1. D and are right angles 1. D 2. D = 2. ll right triangles are congruent 3. D D 3. Transitive ***Theorem 2.4 ongruent Supplements Theorem IF two angles are supplementary to the same angle (or angles), THEN they are congruent. ***Theorem 2.5 ongruent omplements Theorem IF two angles are complementary to the same angle (or angles), THEN they are congruent. Ex. m 1 24, m 3 24, 1 and 2 are complementary, : 3 and 4 are complementary Prove: m 1 24, m and 2 are complementary, 3 and 4 are complementary 2. m 1 = m 3 2. Transitive Property Def. of = angles Substitution ongruent omp. Theorom. ***Linear Pair Postulate IF two angles form a linear pair, Then they are supplementary o 1 2 (over)

13 ***Theorem 2.6 Vertical ngles Theorem Vertical angles are congruent. Ex. : 1 and 2 are a linear pair, 2 and 3 are a linear pair Prove: and 2 are a linear pair 1. 2 and 3 are a linear pair 2. 1 & 2 are supp. 2. Linear Pair Postulate 2 & 3 are supp ongruent Supp. Theorem Ex. See diagram m QD m QE 90 a) m QG 90 (vertical angles) b) m Q 90 ( linear pair) D c) If m QD 31, then m EQF d) If m QG 125, then m QF Q E e) m Q m GQF m EQG f) If m EQF 38, then m Q G F

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