Unit 3 Reasoningand ProofChapter 2 NotesGeometryNameHour1
Unit 3 – Chapter 2 – Reasoning and ProofMonday 2-3 Conditional StatementsSeptember 30Tuesday 2-5 Postulates and ProofDHQ 2-3October 1Block 2-6 Algebraic ProofWed/Thurs.Oct 2/3DHQ 2-5UNO Proof ActivityFriday Quiz 2-3 to 2-6October 4Monday 2-7 Segment ProofOctober 7Tuesday 2-8 Angle ProofOctober 8Block Proof PracticeWed/Thurs.October 9/10FridayOctober 11DHQ 2-7DHQ 2-8No School – Teacher Work DayMonday Proof ReviewOctober 14Tuesday ACT Test TipsOctober 15Wednesday ASVAB/PSAT/ACT PracticeOctober 16Parent Teacher Conferences 4:30-8:30Thursday/Friday No SchoolOctober 17-18Thursday, Parent Teacher Conferences 11-8Monday Proof Review for TestOctober 21Tuesday Proof Review for TestOctober 22Wed/Thurs Proof TestOctober 23/242
2.3 Conditional StatementsI can analyzestatements in ifthen form.Conditional Statement- an if-then statement (𝒑 𝒒)Hypothesis – is the phrase immediately following the word “if”Conclusion – is the phrase immediately following the word “then”I can write theconverse of ifthen statementsExample 2-3-1 Circle the hypothesis and underline the conclusion in thefollowing sentences.a.If a polygon has 6 sides, then it is a hexagon.b.Tamika will advance to the next level of play if she completes the maze in her computergame.c.If today is Thanksgiving Day, then today is Thursday.d.A number is a rational number if it is an integer.Example 2-3-2 Write each statement in if-then form.a) A five-sided polygon is a pentagon.b) An angle that measures 45 is an acute angle.c) An obtuse triangle has exactly one obtuse angle.d)BirdsBlueJaysALERT: A conditional with a false hypothesis is always true.3
Example 2-3-3 Determine the truth value of the conditional statement. If true,explain your reasoning. If false, give a counterexample.If last month was February, thenthis month is March.When a rectangle has an obtuseangle, it is a parallelogram.True/FalseIf Mrs. McWhorter teachesgeometry, then everyone has a“C” in her class.True/FalseIf 2 2 7, then a banana is avegetable.True/FalseIf two angles are acute, thenthey are congruent.True/FalseIf an even number greater than2 is prime, then 5 4 8.True/FalseConverse of aConditionalStatement.True/FalseConverse – The statement formed by the andthe of a conditional statement.Example 2-3-4 Conditional StatementsWrite the conditional and its converse for the following true statement. Determinethe truth value of each statement. If a statement is false, give a counterexample.a. Bats are animals that can fly.Statement (T / F)Converse (T / F)b. The deer population increases with the food available.Statement (T / F)Converse (T / F)4
Biconditional StatementsAll definitions areBiconditionalStatements.Statements whoseconverse is alsotrue, can be writtenas a biconditional.Statements that use the phrase “if and only if” which is abbreviated“iff”Def of segments: Segments are if and only if segments have thesame measure.Def of angles: Angles are if and only if angles have the samemeasure.Def of right angle: an angle is a angle iff the angle’s measure is 90 .Def of segment bisector: a segment/ray bisects a segment iff thesegment/ray goes through theDef of bisector: a segment/ray bisects an iff the segment/ray cuts theperfectly in .Def of Complementary: two angles are complementary iff the sum of theangles is .Def of Supplementary: two angles are supplementary iff the sum of theangles is .5
2.5 Postulates and Paragraph ProofsI can identifyand use basicpostulatesabout points,lines, andplanes.I can writeparagraphproofs.Example 2-5-1 Identifying PostulatesIn the figure, ⃡𝐴𝐶 and ⃡𝐷𝐸 are in plane Q and ⃡𝐴𝐶 ⃡𝐷𝐸 . State whether each postulate istrue or false, then state the postulate that can be used to show each statement is true orfalse.a. Exactly one plane containsb. ⃡𝐵𝐸 lies in plane Q.points F, B, and E.c. Through points F, B, and Gthere is exactly one planed. ⃡𝐴𝐶 and ⃡𝐹𝐺 intersect at B6
Example 2-5-2 Analyze Statements Using PostulatesDetermine whether the following statement is always, sometimes, or never true.Explain.A. If plane T contains ⃡𝐸𝐹 and ⃡𝐸𝐹 contains point G, then plane T contains pointG.B. ⃡𝐺𝐻 contains three noncollinear points.C. There is exactly one plane that contains points A, B, and C.D. Points E and F are contained in exactly one line.E. Two lines intersect in two distinct points M and N.F. The intersection of plane M and plane N is point A.G. If A and B lie in plane W, then AB lies in plane W.H. TR lies in plane M.7
Example 2-5-3 Write a Paragraph Proof⃡ intersects 𝐶𝐷⃡ , write a paragraph proof to show that 𝐴, 𝐶, and 𝐷A. Given 𝐴𝐶determine a plane.It is given ⃡𝐴𝐶 intersects ⃡𝐶𝐷, so they must intersect at point C, by Postulate⃡ and point D is on 𝐶𝐷⃡ . Points A, C, and D are. So Point A is on 𝐴𝐶. Therefore,̅̅̅̅, write a paragraph proof to show that 𝑋𝑀̅̅̅̅̅ B. Given that M is the midpoint of 𝑋𝑌̅̅̅̅̅𝑀𝑌.̅̅̅̅, then from the definition of midpoint of a segment,If M is the midpoint of 𝑋𝑌we know that , This means that and have thesame measure. By the definition of congruent segments, we know that if thesegments have the same measure they are congruent. Therefore, ,Once a conjecture has been proven true, it can be stated as a theorem and used in otherproofs, the conjecture in example 3 is known as the midpoint theorem.Recall:Example 2-5-4 Apply the Midpoint TheoremTransitive propertyIn the figure below, point B is the midpoint of ⃡𝐴𝐶 and point C is the midpoint of ⃡𝐵𝐷.is needed for thisWrite a paragraph proof to prove that AB CD.proof keep ornot?It is given that point B is the midpoint of ⃡𝐴𝐶 and point C is the midpoint of ⃡𝐵𝐷. So,𝐴𝐵 𝐵𝐶 and 𝐵𝐶 𝐵𝐷.8
2.6 Algebraic ProofI can usealgebra to writetwo columnproofs.I can useproperties ofequality to writegeometric proofsProperties of Real Numbersthe following properties are true for any real numbers a, b, and cAddition Property of EqualitySubtraction Property of EqualityMultiplication Property of EqualityDivision Property of EqualityReflexive Property of EqualitySymmetric Property of EqualityTransitive Property of EqualitySubstitution Property of EqualityDistributiveExample 2-6-1 Algebraic ProofJustify each step using a two column proof.Given: 2(5 3𝑎) 4(𝑎 7) 92Prove: 𝑎 11StatementsReasons9
Example 2-6-2 Literal EquationsIf the distance d an object travels is given by 𝑑 20𝑡 5, the time t that the object𝑑 5travels is given by 𝑡 20 . Write a two column proof to verify this conjecture.Begin by stating what is given and what you are to prove.StatementsReasons1. 𝑑 20𝑡 51. Given220.127.116.11.4. 𝑡 4.𝑑 5201If the formula for area of a trapezoid is 𝐴 2 (𝑏1 𝑏2 )ℎ, then what would be theformula in terms of height (h). Fill in the missing statements and reasons in the twocolumn proof.StatementsReasons18.104.22.168.2. Multiplication Property of Equality2𝐴(𝑏1 𝑏2)3. ℎ4.4. Symmetric Property of EqualityExample 2-6-3 Geometric ProofUse a two column proof to verify each conjecture.Given: 𝐴 𝐵, 𝑚 𝐵 2𝑚 𝐶, 𝑚 𝐶 45Prove: 𝑚 𝐴 90Statements1. 𝐴 𝐵, 𝑚 𝐵 2𝑚 𝐶,𝑚 𝐶 452.3. 𝑚 𝐴 2𝑚 𝐶Reasons1.4.4. Substitution5. 𝑚 𝐴 905.̅̅̅̅ 𝐶𝐷,̅̅̅̅̅ 𝐶𝐷̅̅̅̅ ̅̅̅̅Given: 𝐴𝐵𝑅𝑆,𝐴𝐵 12Prove: RS 122. Def of angles3.Statements1.2.3. 𝐴𝐵 𝑅𝑆4.Reasons1.2. Def of segments3.4. Substitution5. 𝑅𝑆 125.10
2.7 Proving Segment RelationshipsI can writeproofs involvingsegmentaddition andsegmentcongruenceIn Chapter 1 we learned about the Segment Addition Postulate. It is used as ajustification in many proofs.In section 2.6, these same properties were introduced as properties of equality, now theycan be used as properties of congruence.Example 2-7-1 Using the Segment Addition Postulate̅̅̅̅, 𝒕𝒉𝒆𝒏 𝑨𝑪̅̅̅̅ 𝑩𝑫̅̅̅̅̅.PROVE that if ̅̅̅̅𝑨𝑩 𝑪𝑫StatementsReasons11
Example 2-7-2: Prove the following̅̅̅̅ ̅̅̅̅Given: 𝑮𝑫𝑩𝑪̅̅̅̅ ̅̅̅̅𝑩𝑪𝑭𝑯̅̅̅̅𝑭𝑯 ̅̅̅̅𝑨𝑬̅̅̅̅ 𝑮𝑫̅̅̅̅Prove: : 𝑨𝑪 𝑨𝑩𝑨𝑩 𝑩𝑿𝑪𝒀 𝑿𝑫Prove: 𝑨𝒀 𝑩𝑫Write a paragraph proof to solve the argument.12
2.8 Proving Angle RelationshipsI can writeproofs involvingsupplementaryandcomplementaryanglesI can writeproofs involvingcongruent andright angles.Example 2-8-1: Using the Angle Addition PostulateGiven: 𝑚 1 23 , 2 is a right angle, 𝑚 𝐴𝐵𝐶 131 Prove: 𝑚 3 18 e 2-8-2Given: QPS TPRProve: 1 3232131StatementsReasons22.214.171.124.126.96.36.199.188.8.131.52.13
Definition of Complementary Angles- Two angles are complementary if and only if thesum of theirDefinition of Supplementary Angles- Two angles are supplementary if and only if thesum of theirDEFINITIONS WORK FORWARDS AND BACKWARDS – ARE BICONDITIONALExample 2-8-3:Given: 1 and 2 are supplementary. 2 and 3 are supplementary.Prove: 1 3Statements1.2.Reasons184.108.40.206.220.127.116.11.Congruent Supplements Theorem:If 5 and 6 are supplementary and 7 and 6 are supplementarythen :If 1 and 4 are complementary and 1 and 2 are complementarythen Linear Pair Theorem:If two angles form a linear pair, then they are supplementary. 1 and 2 form a ,14
Example 2-8-4Given: 1 and 2 form a linear pair. 2 and 3 are supplementary.Prove: 1 3StatementsReasonsa. 1 and 2 form a linear paira.b.b. linear pair theoremc. 2 and 3 are supplementaryc. givend.d.Example 2-8-5 Given: 1 4Prove: 2 3StatementsReasons18.104.22.168.3.3.15
Example 2-8-6:Given: 4 3Prove: 1 2Statements1.Reasons1.2.2. Definition of congruent angles3. 1 and are supplementary 2 and are supplementary3.4.4. Definition of supplementary22.214.171.124.126.96.36.199.Given: 𝑊𝑋𝑌 is a right angle. 𝟏 𝟑Prove: 1 and 2 are 6.6.16
and Proof Chapter 2 Notes Geometry. 2 Unit 3 – Chapter 2 – Reasoning and Proof Monday September 30 2-3 Conditional Statements Tuesday October 1 2-5 Postulates and Proof DHQ 2-3 Block Wed/Thurs. Oct 2/3 2-6 Algebraic Proof DHQ 2-5 UNO Proof Activity
PROOF MARKS: BELGIAN PROOF MARKSPROOF MARKS: BELGIAN PROOF MARKS, cont. BELGIAN PROOF MARKS PrOOF mark CirCa PrOOF hOuse tYPe OF PrOOF and gun since 1852 Liege provisional black powder proof for breech loading guns and rifled barrels - Liege- double proof marking for unfurnished barrels - Liege- triple proof provisional marking for
since 1950 E. German, Suhl choke-bore barrel mark PROOF MARKS: GERMAN PROOF MARKS, cont. PROOF MARKS: ITALIAN PROOF MARKS, cont. ITALIAN PROOF MARKS PrOOF mark CirCa PrOOF hOuse tYPe OF PrOOF and gun since 1951 Brescia provisional proof for all guns since 1951 Gardone provisional proof for all guns
2154 PROOF MARKS: GERMAN PROOF MARKS, cont. PROOF MARK CIRCA PROOF HOUSE TYPE OF PROOF AND GUN since 1950 E. German, Suhl repair proof since 1950 E. German, Suhl 1st black powder proof for smooth bored barrels since 1950 E. German, Suhl inspection mark since 1950 E. German, Suhl choke-bore barrel mark PROOF MARKS: GERMAN PROOF MARKS, cont.
Reasoning and Proof Unit. 2 . 2 4 Deductive Reasoning Law of Detachment If the hypothesis of a true conditional is true, then the conclusion is true. . 2 5 Reasoning in Algebra and Geometry Let's Go Over the Reasons on the Wall!!!! Proof: Given: m ABC 80
Description Logic Reasoning Research Challenges Reasoning with Expressive Description Logics – p. 2/40. Talk Outline Introduction to Description Logics The Semantic Web: Killer App for (DL) Reasoning? Web Ontology Languages DAML OIL Language Reasoning with DAML OIL OilEd Demo Description Logic Reasoning Research Challenges Reasoning with Expressive Description Logics – p. 2/40. Talk .
Use inductive reasoning to make a conjecture about the sum of a number and itself. Then use deductive reasoning to show that the conjecture is true. 11. Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. All multiples of 8 are divisible by 4. 64 is a multiple of 8. So, 64 is .
(Henningsen, 1997). The chapters examine were Chapter 2- Reasoning and Proof, Chapter 3- Parallel and perpendicular lines and Chapter 4 – Congruent Triangles. Chapter 2, entitled “Reasoning and proof ”, consists of five sections: Conditional statements, Biconditionals and Definitions, Deductive Reasoning, Reasoning in
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