Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Name: ________________________ May, 2015 Conditional Statements Conclusion Converse statements Inverse statements Contrapositive statements Contrapositive statements Statement: Μ Μ Μ Μ πΆπ΄ β Μ Μ Μ Μ πΆπ΄ Reason: Reflexive Property Statement: β π·ππ΄ β β πΆππ΅ Reason: Vertical angles are congruent Proofs and Logic IP Castoro / Cheung Proofs and Logic Independent Practice Logic How-To If a, then b. If a. Negation: to deny Then b. If b, then a. If not a, then not b. Counter-Example: an example to disprove a statement If not b, then not a. Most Common Proof Moves Μ Μ Μ Μ Μ Statement: β π΄π΅π· β Statement: β π΄π·πΆ β β πΆπ·π΅ Statement: π΄π β Μ Μ Μ Μ Μ ππ΅ Reason: Right angles are β π·π΅πΆ Reason: Definition of Reason: Definition of always congruent midpoint angle bisector Statement: β π β β π Reason: Alternate interior angles are congruent Statement: β π β β πΌ Reason: corresponding angles are congruent SSS Congruence Postulate Congruent Triangles SAS Congruence Postulate AAS Congruence Postulate HL Congruence AA Similarity Postulate SSS Similarity Similar Triangles SAS Similarity Statement: β π·πΆπ΄, β π·πΆπ΅ are right angles Reason: Definition of perpendicular lines ASA Congruence Postulate CPCTC β corresponding parts of congruent triangles are congruent Corresponding sides of similar triangles are proportional The product of the means is equal to the product of the extremes Name: ________________________ May, 2015 Proofs and Logic IP Castoro / Cheung Logic Practice 1. Which compound statement is true? 2. A student wrote the sentence β4 is an odd integer.β 1) A triangle has three sides and a quadrilateral What is the negation of this sentence and the truth value has five sides. of the negation? 2) A triangle has three sides if and only if a 1) 3 is an odd integer; true quadrilateral has five sides. 2) 4 is not an odd integer; true 3) If a triangle has three sides, then a quadrilateral 3) 4 is not an even integer; false has five sides. 4) 4 is an even integer; false 4) A triangle has three sides or a quadrilateral has five sides. 3. What is the inverse of the statement βIf two triangles 4. What is the converse of the statement "If , are not similar, their corresponding angles are not then is a right triangle"? congruentβ? 1) If is a right triangle, then . 1) If two triangles are similar, their corresponding 2) if, and only if, is a right angles are not congruent. triangle. 2) If corresponding angles of two triangles are not 3) If is not a right triangle, then congruent, the triangles are not similar. . 3) If two triangles are similar, their corresponding 4) If angles are congruent. , then is not a right 4) If corresponding angles of two triangles are triangle. congruent, the triangles are similar. 6. What is the contrapositive of the statement, βIf I am tall, 5. If , which statement is false? 1) x is prime and x is odd. then I will bump my headβ? 2) x is odd or x is even. 1) If I bump my head, then I am tall. 3) x is not prime and x is odd. 2) If I do not bump my head, then I am tall. 4) x is odd and 2x is even. 3) If I am tall, then I will not bump my head. 4) If I do not bump my head, then I am not tall. 7. Given the statement: βIf two lines are cut by a 8. Given the statement, "If a number has exactly two transversal so that the corresponding angles are factors, it is a prime number," what is the contrapositive of congruent, then the lines are parallel.β What is true about this statement? the statement and its converse? 1) If a number does not have exactly two factors, 1) The statement and its converse are both true. then it is not a prime number. 2) The statement and its converse are both false. 2) If a number is not a prime number, then it does 3) The statement is true, but its converse is false. not have exactly two factors. 4) The statement is false, but its converse is true. 3) If a number is a prime number, then it has exactly two factors. 4) A number is a prime number if it has exactly two factors. Complete the statements below and state whether they are true or false: If two numbers are even, then their sum is even. Hypothesis:__________________________________________________________________________ Conclusion:__________________________________________________________________________ Converse:___________________________________________________________________________ Inverse: _____________________________________________________________________________ Contrapositive:_______________________________________________________________________ Name: ________________________ May, 2015 Proofs and Logic IP Castoro / Cheung Proof Practice State the postulate that proves the following triangles congruent or similar from the word bank below: D. SAS A. SSS B. AAS C. ASA E. SAS F. AA Similarity G. HL Similarity 1. 2. 3. 4. 5. 6. 7. Match the proof move to the appropriate diagram: A. Vertical Angles 1. H. SSS Similarity 8. 2. 3. 5. 6. B. Reflexive Property C. Corresponding Angles D. Angle Bisector E. Alternate Interior Angles F. Right Angles 1. In the diagram of , and 4. and . Which method can be used to prove 1) SSS 2) SAS 3) ASA 4) HL Proof Multiple Choice Practice 2. In the accompanying diagram of triangles BAT and FLU, below, , and . ? Which statement is needed to prove 1) 2) 3) 4) ? Name: ________________________ May, 2015 Proofs and Logic IP Castoro / Cheung 3. In the diagram below, four pairs of triangles are shown. Congruent corresponding parts are labeled in each pair. Using only the information given in the diagrams, which pair of triangles can not be proven congruent? 1) A 2) B 3) C 4) D 5. Which condition does not prove that two triangles are congruent? 1) 2) 3) 4) 7. In the diagram below of and intersect at E, such that and , and . Triangle DAE can be proved congruent to triangle BCE by 1) ASA 2) SAS 3) SSS 4) HL 4. Given that ABCD is a parallelogram, a student wrote the proof below to show that a pair of its opposite angles are congruent. What is the reason justifying that ? 1) Opposite angles in a quadrilateral are congruent. 2) Parallel lines have congruent corresponding angles. 3) Corresponding parts of congruent triangles are congruent. 4) Alternate interior angles in congruent triangles are congruent. 6. Which statements could be used to prove that and are congruent? 1) 2) 3) 4) 8. As shown in the diagram below, . Which method could be used to prove 1) SSS 2) AAA 3) SAS 4) AAS bisects and ? Name: ________________________ May, 2015 9. In the diagram below of point on , Proofs and Logic IP Castoro / Cheung , Q is a point on is drawn, and , S is a below, . Which reasons can be used to prove ? 1) reflexive property and addition postulate 2) reflexive property and subtraction postulate 3) transitive property and addition postulate 4) transitive property and subtraction postulate and , information would prove 1) 2) 3) 4) Which statement must be true? 1) 2) 3) 4) 12. In the diagram below, . Which statement is not always true? 1) 2) 3) 4) Which statement must be true? 1) 2) 3) 4) 15. In . . Which reason justifies the conclusion that ? 1) AA 2) ASA 3) SAS 4) SSS 11. The diagram below shows a pair of congruent triangles, with and . 13. In the diagram of 10. In the diagram below, . Which additional ? 14. When writing a geometric proof, which angle relationship could be used alone to justify that two angles are congruent? 1) supplementary angles 2) linear pair of angles 3) adjacent angles 4) vertical angles 16. In triangles ABC and DEF, , , , , and . Which method could be used to prove ? 1) 2) 3) 4) AA SAS SSS ASA Name: ________________________ May, 2015 Given: Μ Μ Μ Μ πΏπ bisects Μ Μ Μ Μ Μ ππ at N β π β β π Prove: π₯πΏππ β π₯πππ Statement Μ Μ Μ Μ bisects ππ Μ Μ Μ Μ Μ atCN πΏπ E β π β β π Μ Μ Μ Μ πΏπ β Μ Μ Μ Μ ππ β πΏππ β β πππ π₯πΏππ β π₯πππ B Proofs and Logic IP Castoro / Cheung Reason Given Given Definition of bisector Vertical angles are congruent ASA Theorem Why is the circled statement wrong? What is the correct statement? D Given: Μ Μ Μ Μ π΄π· bisects Μ Μ Μ Μ π΅πΆ at E, Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ π΅πΆ , Μ Μ Μ Μ π·πΆ β Μ Μ Μ Μ π΅πΆ Μ Μ Μ Μ β π·πΆ Μ Μ Μ Μ Prove: π΄π΅ Given: Μ Μ Μ Μ π΄π΅ || Μ Μ Μ Μ π·πΈ Μ Μ Μ Μ π·πΆ Prove: Μ Μ Μ Μ π΅πΆ Μ Μ Μ Μ π·πΈ = Μ Μ Μ Μ π΅π΄ Μ Μ Μ Μ β π΅πΆ Μ Μ Μ Μ , π΅π· Μ Μ Μ Μ Μ Μ Μ Μ bisects line π΄πΆ Given: π΄π΅ Prove: βπ΄π΅π· β βπΆπ΅π· Name: ________________________ May, 2015 1. 4 2. 2 1. B 2. F 1. B 3. 3 3. D 2. C 1. 3 9. 1 2. 1 10. 2 Proofs and Logic IP Castoro / Cheung Logic Multiple Choice Answer Key: 4. 2 5. 3 6. 4 Matching Postulates 4. H 5. E 3. F 3. 1 11. 4 Matching Proof Moves 4. D 7. 1 6. G 7. A 8. C 5. A Proof Multiple Choice Answer Key: 4. 3 5. 2 6. 2 12. 4 13. 2 14. 4 8. 2 6. E 7. 1 15. 3 8. 4 16. 2 Proof Answer Key Why is the circled statement wrong? What is the correct statement? Μ Μ Μ Μ bisects ππ Μ Μ Μ Μ Μ . The correct statement is: Μ Μ Μ Μ Μ Μ Μ Μ Μ The circled statement is incorrect because πΏπ ππ β ππ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Given: π΄π· bisects π΅πΆ at E, π΄π΅ β π΅πΆ , π·πΆ β π΅πΆ Μ Μ Μ Μ β π·πΆ Μ Μ Μ Μ Prove: π΄π΅ Statement Reason Μ Μ Μ Μ bisects π΅πΆ Μ Μ Μ Μ at E, π΄π΅ Μ Μ Μ Μ β π΅πΆ Μ Μ Μ Μ , π·πΆ Μ Μ Μ Μ β π΅πΆ Μ Μ Μ Μ Given π΄π· Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Definition of bisector Μ Μ Μ Μ π΅πΈ β πΈπΆ β ABE and β ECD are right angles Definition of perpendicular lines β ABE β β ECD All right angles are congruent β AEB β β CED Vertical angles are congruent ASA Congruence βπ΄πΈπ΅ β βπ·πΈπΆ Μ Μ Μ Μ β π·πΆ Μ Μ Μ Μ CPCTC π΄π΅ Given: Μ Μ Μ Μ π΄π΅ || Μ Μ Μ Μ π·πΈ Μ Μ Μ Μ π«πͺ Prove: Μ Μ Μ Μ π©πͺ Μ Μ Μ Μ π«π¬ = Μ Μ Μ Μ π©π¨ Statement Μ Μ Μ Μ || π·πΈ Μ Μ Μ Μ π΄π΅ β DCE β β ACB β EDC β β CBA βπ·πΈπΆ~βπ΄π΅πΆ Μ Μ Μ Μ Μ Μ Μ Μ π·πΆ π·πΈ = Μ Μ Μ Μ Μ Μ Μ Μ π΅πΆ π΅π΄ Given: Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ π΅πΆ , Μ Μ Μ Μ π΅π· bisects line Μ Μ Μ Μ π΄πΆ Prove: βπ΄π΅π· β βπΆπ΅π· Reason Given Vertical angles are congruent Alternate interior angles are congruent AA Similarity Corresponding parts of similar triangles are proportional Statement Μ Μ Μ Μ Μ Μ Μ Μ π΄π΅ β π΅πΆ , Μ Μ Μ Μ π΅π· bisects line Μ Μ Μ Μ π΄πΆ Μ Μ Μ Μ β π·πΆ Μ Μ Μ Μ π΄π· Μ Μ Μ Μ π΅π· β Μ Μ Μ Μ π΅π· βπ΄π΅π· β βπΆπ΅π· Reason Given Definition of line bisector Reflexive property SSS Congruence Name: ________________________ May, 2015 Proofs and Logic IP Castoro / Cheung Exit Ticket Which statement is logically equivalent to the statement "If Corey worked last summer, he buys a car"? 1) If Corey does not buy a car, he did not work last summer. 2) If Corey buys a car, he worked last summer. 3) If Corey did not work last summer, he does not buy a car. 4) If Corey buys a car, he did not work last summer. In the accompanying diagram, bisects and . What is the most direct method of proof that could be used to prove 1) 2) 3) 4) Given: Prove: and intersect at B, , and bisects . ?