Chapter 2 Notes Review PDF

2.2 Extended Notes Determine the Truth-Value of the following conjectures. If it is False provide a counter-example. 1. Given: m 1 = 20 and m 2 = 70 2. If points A, B, and C are collinear, then pt. B is Conjecture: 1 and 2 are comp. ’s. be...

2.2 Extended Notes Determine the Truth-Value of the following conjectures. If it is False provide a counter-example. 1. Given: m 1 = 20 and m 2 = 70 2. If points A, B, and C are collinear, then pt. B is Conjecture: 1 and 2 are comp. ’s. between pts. A and C. 3. Given: 1 and 2 are sup. ’s 4. If a quadrilateral has four sides, then it is a Conjecture: 1 2 square. *Def.: quadrilateral – a four-sided figure 5. If line m is to line n then four right angles 6. Given: A is an obtuse angle are formed. Conjecture: m A = 120 *Symbol: - Perpendicular 7. If AB CD and CD EF then AB EF 8. If a polyhedron is a prism, then its bases are triangles. *Def.: polyhedron – many sided solid figure Example: 9. Given: M is the midpoint of AB 10. If two angles are comp. then a ray bisects a right angle. Conjecture: AM MB  2.3 Extended Notes 1. Determine if the conjecture is TRUE or FALSE based on the given information. If false, provide a counterexample. Given: ⃡𝑋𝑌 and ⃡𝑍𝑊 intersect at A Conjecture: ZAX and WAY are vertical ’s 2. The following statement is the CONTRAPOSITIVE of a given conditional. Write the converse AND determine the truth-value of the converse. If 2 ’s are not adj. then they are not a linear pair. 3. The following statement is the INVERSE of a given conditional. Write the converse AND determine the truth-value of the converse. If 2 ’s are not vertical ’s then they are not ≅.  2.4 Extended Notes Test the statements to see if it is TRUE and if its converse is TRUE. If so, write it as a biconditional. If not, write CANNOT FORM A BICONDITIONAL. 1. If 2 ’s are vertical ’s then they are ≅. 2. If 2 ’s are a linear pair then they are adj. and supp. 3. If a quadrilateral is a square, then it has four sides and four ’s. 4. If a ray bisects a 90° then comp. ’s are formed. 5. If 2 rays are opposite rays, then they share a common endpoint. 6. If AB ≅ CD and CD ≅ EF then AB ≅ EF. 2.5 Extended Notes Lewis Carroll, author of Alice in Wonderland, was also a famous logician. Some of his most famous logic puzzles involved non-sense statements that have a logical conclusion based on the Law of Syllogism. Below are some of such puzzles. Find the logical conclusion based on the following statements. (Hint: keep in mind the contrapositive has the same truth-value as the conditional) 1. (a) All babies are illogical. (b) Nobody is despised who can manage a crocodile. (c) Illogical persons are despised. 2. (a) None of the unnoticed things, met with at sea, are mermaids. (b) Things entered in the log, as met with at sea, are sure to be worth remembering. (c) I have never met with anything worth remembering, when on a voyage. (d) Things met with at sea, that are noticed, are sure to be recorded in the log. 2.5 Extended Notes 3. (a) No interesting poems are unpopular among people of real taste. (b) No modern poetry is free from affectation. (c) All your poems are on the subject of soap-bubbles. (d) No affected poetry is popular among people of real taste. (e) No ancient poem is on the subject of soap-bubbles. 4. (a) All writers, who understand human nature, are clever. (b) No one is a true poet unless he can stir the hearts of men. (c) Shakespeare wrote “Hamlet”. (d) No writer, who does not understand human nature, can stir the hearts of men. (e) None but a true poet could have written “Hamlet”. 2.5 Extended Notes 5. (a) Promise breakers are untrustworthy. (b) Wine drinkers are very communicative. (c) A man who keeps his promises is honest. (d) No teetotalers are pawnbrokers. (e) One can always trust a very communicative person. 6. (a) I despise anything that cannot be used as a bridge. (b) Everything, that is worth writing an ode to, would be a welcome gift to me. (c) A rainbow will not bear the weight of a wheelbarrow. (d) Whatever can be used as a bridge will bear the weight of a wheelbarrow. (e) I would not take, as a gift, a thing that I despise.  2.6 Extended Notes Write a 2-column proof. 1. Given: 12 − 3(2𝑤 + 1) = 7𝑤 − 3(7 + 𝑤) Prove: 𝑤 = 3  2.6 Extended Notes Write a 2-column proof. 2. Given: 7(𝑎 + 1) − 3𝑎 = 5 + 4(2𝑎 − 1) 3 Prove: 𝑎 = 2  2.6 Extended Notes Write a 2-column proof. 3. Given: 3(𝑥 − 2) + 𝑥 = 2𝑥 + 12 Prove: 𝑥 = 9  2.7 Extended Notes Write a 2-column proof. 1. Given: 𝐴𝐵 ≅ 𝐶𝐷 A B C D Prove: 𝐴𝐶 ≅ 𝐵𝐷  2.7 Extended Notes Write a 2-column proof. 2. Given: 1 is supp. to 2; 3 is supp. to 2 Prove: 1 3 1 2 3  2.7 Extended Notes Write a 2-column proof. 3. Given: ⃗⃗⃗⃗⃗ 𝐸𝐶 bisects ∠𝐴𝐸𝐷; ⃗⃗⃗⃗⃗ 𝐸𝐷 bisects ∠𝐶𝐸𝐵 Prove: ∠1 ≅ ∠2 C D 3 1 2 A E B  2.7 Extended Notes Write a 2-column proof. 4. Given: 𝐴𝐵 ≅ 𝐶𝐷; M is the midpt. of 𝐴𝐵; N is the midpt. of 𝐶𝐷 Prove: 𝐴𝑀 ≅ 𝑁𝐷 A M B C N D  2.7 Extended Notes Write a 2-column proof. 5. Given: 𝐴𝐵 ≅ 𝐶𝐷; 𝐵𝐷 ≅ 𝐷𝐸 Prove: 𝐴𝐷 ≅ 𝐶𝐸 C A B D E  2.7 Extended Notes Write a 2-column proof. 6. Given: C is the midpt. of 𝐴𝐵, 𝐷𝐸 ≅ 𝐴𝐶 Prove: 𝐶𝐵 ≅ 𝐷𝐸 A C B D E  2.7 Extended Notes l Write a 2-column proof. 7. Given: Line l bisects 𝑋𝑌at pt. W; D is the midpt. of 𝐶𝐸; 𝑊𝑌 = 𝐶𝐷 X W Y Prove: 𝑋𝑊 ≅ 𝐷𝐸 C D E  2.7 Extended Notes Write a 2-column proof. 8. Given: 𝐴𝐵 ≅ 𝑋𝑊, 𝐴𝐵 ≅ 𝑊𝑌 X W Y Prove: W is the midpt. of 𝑋𝑌 A B