`Key ConceptsReOexive Property Symmetric Property Transitive PropertyProperties of Congruence AB ~ AB LA~LA If AB ~ CD, then CD ~ AB. If LA ~ LB, then LB ~ LA. If AB ~ CD and CD ~ EF, then AB ~ EF. If LA ~ LB and LB ~ LC, then LA ~ LC.Using Properties of Equality and CongruenceName the property of equality or congruence that justifies each statement. a. LK ~ LK Reflexive Property of Congruenceb. If2x - 8= 10, then 2x = 18. Addition Property of Equalityc. If RS ~ TW and TW ~ PQ, then RS ~ PQ. Transitive Property of Congruenced. If mLA = mLB, then mLB = mLA.Symmetric Property of Equalityri! Quick Check eEXERCISESName the property of equality or congruence illustrated. a.XY~XY b. If mLA = 45 and 45 = mLB , then mLA = mLB.For more exercises, see Extra Skill, Word Problem, and Proof Practice.oPractice by Example Examples 1 and 2~AlgebraFill in the reason that justifies each step.1. Solve for x. mLCDE + mLEDFfor Help(page 104)= 180a..L:x + (3x + 20) = 180 4x + 20 = 180 4xxb. ~c.~= 160 = 40d.~e.~3(n2. Solve for n. Given: XY3(n+ 4)= 42XYa.~x b.~c.~z·3nyXZ.+ ZY=+ 4) + 3n = 42 3n + 12 + 3n = 42 6n + 12 = 426nd.~e._~ f.~lesson 2-4 Reasoning in Algebra=30n =5'&quot;105~Algebra 3.Give a reason for each step.1x - 5 = 10z(!x x -Given4. 5(x= 20 10 = 20 x = 305)a· .-:L b.~ c.~+ 3) = -4 5x + 15 = -45x = -19 x = _195Givena.~b.~c. ....LExample 3(page 105)Name the propertythat jnstifies each statement.5. LZ==LZ6. 2(3x+ 5) = 6x +107. If 12x=84, then x==7.8. If ST== QR,then QR== ST.9. If mLA 11. If 3x15, then 3mLA= =45.10. XY=XY12. If KL+14=80, then 3x y,66.= MN, then MN = KL.13. If 2xthen+ Y = 5 and x = 2x + x = 5.14. IfAB - BC = 12, then AB = 12 + Be.o15. If Ll Apply Your Skills== L2and L2==L3, then L1== L3.17. Subtraction Property of Equality If 5x + 6 = 21, then Z. = 15.19. Symmetric Property of Congruence If LH == LK, then ....L == LH. 21. Distributive Property 3(x - 1) = 3x - ....L 23. Transitive Property of Congruence If LXYZ == LAO Band LAOB == L WfT, then....L. is equivalent 180 to the left side of thisUse the given propertyto completeeach statement.16. Addition Property of Equality If 2x - 5 = 10, then 2x = ....L.18. Symmetric Property of Equality If AB = YU, then ....L. 20. Reflexive PropertyLPQR==....Lof Congruence22. Substitution Property If LM = 7 and EF + LM then....L = NP.= NP,24. Multiple Choice Which expression equation?-4x+ 7y + t(12x - 3y) = CD 6y + 8® 8x + 4y~ ~CO 6y® 8x~&quot;nline Homework Video TutorVisit: PHSchool.com25. Writing Jero claims that the statements LR == RL and LCBA == LABC are both true by the Reflexive Property of Congruence. Explain why Jero is correct. 26. Use what you know about transitive properties The Transitive Property of Falling Dominoes: to complete the following:Web Code: aue-0204If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino ....L to fall.106Chapter 2Reasoning and Proof~27. AlgebraFill in the reason that justifies each step.4xGiven: C is the midpoint of AD.C is the midpoint of AD. AC= CD 4x = Ix + 12 2x = 12 x=6 ~For a guide to solving Exercise 28, see p. 109.a. ?A· C2x+ 12· Db.~c.~d.~e.~28. Algebra In the figure at the right, KM a. Solve for x. Justify each step. b. Find the length of KL.= 35.K.2x- 5.L2xMbd 29. Algebra~In the figure at the right, mLGFI a. Solve for x. Justify each step. b. Find mLEFI.=128.GF I30. ---=+ Algebra Point C is on the crease when you fold ~ BD onto BA. Give the reason that justifies each step. (Hint: See page 102, Exercises 4 and 5.)--BC bisects LABD. mLABC = mLCBD 6n + 1 2na. -.L b.-.L(6&quot; +1)~&quot;.C(4&quot; + 19)0D= 4n + 19 = 18«.s:d.~an=9Challenge 31. Error Analysis Given: a-:»:The steps below &quot;show&quot; that 1= 2. Find the error.=ba=b ab = b2 ab - a2 = b2 - a2 a(b - a) = (b + a)(b - a) a=b+a a=a+aa 1=Given Multiplication Property of Equality Subtraction Property of Equality Distributive Property Division Property of Equality Substitution Property Simplify. Division Property of Equality2a 2=Relationships The relationships &quot;is equal to&quot; and &quot;is congruent to&quot; are reflexive, symmetric, and transitive. In a later chapter, you will see that this is' also trne for the relationship &quot;is similar to.&quot; Consider the foUowing relationships among people. State whether each relationship is reflexive, symmetric, transitive, or none of these. Sample: The relationship &quot;is younger than&quot; is transitive. If Sue is younger than Fred and Fred is younger than Alana, then Sue is younger than Alana. The relationship &quot;is younger than&quot; is not reflexive because Sue is not younger than herself. It is also not symmetric because if Sue is younger than Fred, Fred is not younger than Sue.Real-World8ConnectionPresident Calvin Coolidge, advice columnist Ann Landers, and musician BillWithers · were all born on the Fourth of July. Each one of them &quot;has the same birthday as&quot; ,~·either one of the others.32. has the same birthday as 34. lives in the same state as 36. is the same height as33. is taller than 35. lives in a different state than 37. is a descendant of~ nlinelesson quiz,PHSchool.com, Web Code: aua-0204Lesson 2-4Reasoning in Algebra107Multiple Choice38. Which property justifies this statement? If 4x = 16, then 16 = 4x. A. Multiplication Property of Equality B. Transitive Property of Equality C Reflexive Property of Equality D. Symmetric Property of Equality 39. The Multiplication 3 F. If Lfx H.lf=Property of Equality justifies which statement below?ix6, then&quot;&quot;43x=6.3 G. If Lfx+ 5 - 6, then Lfx - 1._3_= 6, then 3x = 24.J. If ix - 18 = 6, then ix = 24.40. A transitive property justifies which statement below? A.lfy17 = g,theny= 9 + 17. B. If AM = RS, then RS = AM. Clf 5(3a - 4) = 120, then 15a - 20 = 120. D. If LJ = LR and LR = LH, then LJ = LH. 41. Which equation follows from ~m of Equality? F. m Short Response+ 1 = 10 by the Multiplication PropertyH. 1 3m - 9 = 0 J. m - 27+3=30G.1m=9= 042. In the diagram, x = 2y + 15 and x + y = 120. a. Use a Property of Equality to explain why 3y + 15 = 120. b. Solve for y. Justify each step. Then find the value of x.14. for~HelpLesson 2-3Reasoning Use logical reasoning to draw a conclusion.43. If a student is having difficulty in class, then that student's Elena is having difficulty in history class. teacher is concerned.44. If a person has a job, then that person is earning money. If a person is earning money, then that person can save money each week.Lesson 1-6Use the diagram at the right and find each measure.45. mLAOC 47. mLDOB46.mLAOD 48.mLBOEA49. In the diagram, name an obtuse angle and a right angle.lesson 1-1Find the next two terms in each sequence.50. 19,21.5,24,26.5 52. -2,6, -18,54 51. 3.4,3.45,3.456,3.4567 53.8, -4,2,-1108Chapter 2Reasoning and ProofProof ~~~Proving Theorem 2.,2. Study what you are given, what you are to prove, and the diagram. Write a paragraph proof.Given: Prove: Ll and L2 are supplementary.L3 and L2 are supplementary.L1==L3~Proof: By the definition of supplementary angles, mL1 + mL2 = 180 and mL3 + mL2 = 180. By substitution, mLl + mL2 = mL3 + mL2. Subtract mL2 from each side. You get mLl = mL3, or Ll == L3.@Quick CheckeIn the proof above, which Property of Equality allows you to subtract mL2 from each side of the equation?Theorem 2-3 is like the Congruent Supplements Theorem. You can demonstrate its proof in Exercises 7 and 28.Key ConceptsCongruent Complements TheoremIf two angles are congruent and supplementary, then each is a right angle.You can complete proofs of Theorems 2-4 and 2-5 in Exercises 14 and 21, respectively.EXERCISESFor more exercises, see Extra Skill, Word Problem, and Proof Practice.ePractice by Example Example 1(page 111)Find the value of each variable.1.(80 - x)?2.3.I~for~HelpFind the measures of the labeled angles in each exercise. 4. Exercise 1 5. Exercise 2 6. Exercise 3112Chantor 7R.,,,,cnninn &quot;&quot;rl Dr&quot;,,-/'Example 2(page 112)7. OevelopingProofComplete this proof of one form of Theorem 2-3 by filling inthe blanks. If two angles are complements of the same angle, then the two angles are congruent.Given: L1 and L2 are complementary.L3 and L2 are complementary.Prove: L1== L3Proof: By the definition of complementary angles,omL1 + mL2 = a.~andmL3 + mL2 = b.~. Then mL1 + mL2 = mL3 + mL2 by c.~. Subtract mL2 from each side. You get mL1 = d.~,Apply Your Skills ~L/ ~Lor L1== L3.8. WritingHow is a theorem different from a postulate? Give an example of vertical angles in your home.9. Open-Ended10. Reasoning Explain why this statement is true: If mL1 + mL2 = 180 and mL3 + mL2 = 180, then L1== L3.11. Design The two back legs of the director's chair pictured at the left meet in a 72° angle. Find the measure of each angle formed by the two back legs. ~ Algebra Find the value of each variable and the measure of each labeled angle.tz,13.(3x+ Sr(5x(5x - 20)°+ 4yr14. Developing Proof Complete this proof of Theorem 2-4 by filling in the blanks. All right angles are congruent.Given: LX and L Yare right angles. Prove: LXExercise 11== L YvLX YPark St.By the definition of a. ~, mLX = 90 and mL Y = 90.. By the Substitution Property, mLX = b.~, or LX == L YProof:15. Multiple Choice What is the measure of the angle formed by Park St. and Oak St.?® 35°CD55°® 45° ® 90°Name two pairs of congruent angles in each figure. Justify your answers.~.nlineHomework Video TutorVisit: PHSchool.comWeb Code: aue-020S16.B 17'~EI18.KFG H19. Coordinate Geometry LDOE contains points D(2, 3),0(0,0), and E(5, 1). ~ _fmJ:Lthe coordinates of a pointFso that OF is a sideofanangle that is 'adjacent and supplementary to LDOE.lesson 2·5 Proving Angles Congruent11320. Coordinate Geometry LAOX contains points A(l, 3),0(0,0), and X( 4,0). a. Find the coordinates of a-pointB so that LBOA and LAOX are adjacent complementary angles. ----'&gt; b. Find the coordinates of a point C so that OC is a side of a different angle that is adjacent and complementary to LAOX. 21. Developing Proof Complete this proof of Theorem 2-5 by filling in the blanks. If two angles are congruent and supplementary, then each is a right angle. Given: L Wand LV are congruent and supplementary. Prove: L Wand L V are right angles.L\ ~\:&gt;/'V=WProof: L Wand L V are congruent, so mL W = mL a. ...L.. L Wand L V are supplementary so mL W + mL V = b· ...L.. Substituting mL W for mL V, you get mL W + mL W By the c· ...L. Property of Equality, mL W = 90. Since L W 180, or 2mL W=180.== LV, mL V = 90, too. Then both angles are d. ...L. angles.@\Exercise 2222. Sports In the photograph, the wheels of the racing wheelchair are tilted so that L1 == L2. What theorem can you use to justify the statement L3 == L4? ~ Algebra Find the measure of each angle.23. LA is twice as large as its complement, LB.24. LA is half as large as its complement, LB.25. LA is twice as large as its supplement, LB. 26. LA is half as large as twice its supplement, LB.j!QQf. 27. Write a proof for this form of Theorem 2-2.If two angles are supplements of congruent angles, then the two angles are congruent. Given: L1 and L2 are supplementary. L3 and L4 are supplementary. L2 == L4Prove: Ll == L3 j!QQf. 28. Write a proof for this form of Theorem 2-3.If two angles are complements of congruent angles, then the two angles are congruent. Given: L1 and L2 are complementary. L3 and L4 are complementary. L2 == L4a114Prove: Ll == L3Challenge 29. Paper Folding After you've done the Activity on page 110, answer these questions. a. How is the first fold line you make related to angles 3 and 4? b. How is the second fold line you make related to angles 1 and 2? c. How are the two fold lines related to each other? Give a convincing argument to support your answer.Chapter 2Reasoning and Proof~Algebra Find the value of each variable and the measure of each labeled angle. 30. 31. (x+ Y + 5t32.Gridded ResponseFind the measure of each angle. 33. an angle with measure 8 lessthan the measure of its complement 34. one angle of a pair of complementary vertical angles 35. an angle with measure three times the measure of its supplement Use the diagram at the right to find the measure of each of the following angles. 36. L 1 38. L3 37. L2 39. L4 4 3 270° 1lesson 2~4Use the given propertyto completeeach statement.40. Subtraction Property of Equality If 3x + 7 = 19, then 3x = ~. 41. Reflexive Property AB=~ 42. Substitution Property If MN = 3 and MN lesson 2-3 Use deductive reasoning of Congruence+ NP = 15, then~.to draw a conclusion. If not possible, write not possible.43. If two lines intersect, then they are coplanar. Lines m and n are coplanar. 44. If two angles are vertical angles, then they are congruent. L1 and L2 are vertical angles. Lesson 2-2 Each conditional statement below is true. Write its converse. If the converse is also true, combine the statements as a biconditional. 45. Ify+7=32, theny=25.46. If you live in Australia, 47. Ifnthen you live south of the equator.&gt; 0, thenn2 &gt; 0.&quot;inlinelesson quiz, PHSchool.com, Web Code: aua-0205lesson 2-5Proving Angles Congruent115Chapter Review.,5Jjbiconditional (p. 87) conclusion (p. 80) conditional (p. 80) converse (p. 81) deductive reasoning (p. 94) hypothesis (p. 80) law of Detachment (p. 94) law of Syllogism (p. 95) paragraph proof (p. 111) Reflexive Property (p. 105) Symmetric Property (p. 105) theorem (p. 110) Transitive Property (p. 105) truth value (p. 81)Choose the correct vocabulary term to complete each seutence. 1. The statement&quot; LA ~ LA&quot; is an example of the ~ Property of Congruence.2. In a conditional statement, the part that directly follows if is the ~. 3. &quot;If LA ~ LB and LB ~ LC, then LA ~ LC&quot; is an example of the ~ Property of Congruence. 4. When a conditional and its converse are true, they may be written as a single true statement called a ~. 5. The ~ of a conditional switches the hypothesis and the conclusion. Property6. &quot;If LA ~ LB, then LB ~ LA&quot; is an example of the ~ of Congruence.rGo '. nlinePHSchool.com7. The part of a conditional statement that follows &quot;then&quot; is the ~. 8. A conditional has a ~ of true or false.For: Vocabulary quiz Web Code: auj-02S19. Reasoning logically from given statements to a conclusion is ~. 10. A statement that you prove true is a .L.Ski lis and Conce~ts2·1 and 2·2 Objectives&quot; To recognize conditional statements To write converses of conditional statementsAn if-then statement is a conditional. The part following ifis the hypothesis. The part following then is the conclusion. You find the truth value of a conditional by determining whether it is true or false. The symbolic form of a conditional is p --+ q. The converse of a conditional switches the hypothesis and the conclusion. The symbolic form of the converse of p --+ q is q --+ p. When a conditional and its converse are true, you can combine them as a true biconditional. To write a biconditional, you join the two parts of each conditional with the phrase if and only if The symbolic form of a biconditional is p -H q. For Exercises 11-13, (a) write the converse and (b) determine the truth value of the conditional aad its converse. (c) If both statements are true, write a biconditional. 11. If you are a teenager, then you are younger than 20. U. If an angle is obtuse, then its measure is greater than 90 and less than 180. 13. If a figure is a square, then it has four sides.VY To write biconditionals &quot; To recognize good definitionsChapter 2Chapter Review11714. Write the following sentence as a conditional: All flowers are beautiful. A good definition is precise. A good definition uses terms that have been previously defined or are commonly accepted. 15. Rico defines a book as something you read. Explain why this is not a good definition. 16. Write this definition as a biconditional: An oxymoron is a phrase that contains contradictory terms. 17. Write this biconditional as two statements, a conditional and its converse: Two angles are complementary if and only if the sum of their measures is 90.2-3 ObjectivesV To use the Law of Detachment V To use the Law of SyllogismDeductive reasoning is the process of reasoning logically from given statements to a conclusion. If the given statements are true, deductive reasoning produces a true conclusion. The following are two important laws of deductive reasoning: Law of Detachment: If p ~ q is a true statement and p is true, then q is true. Law of Syllogism: If p ~ q and q ~ r are true statements, then p ~ r is true. Use the Law of Detachment to make a conclusion. 18. If you practice table tennis every day, you will become a better player. Lucy practices table tennis every day. 19. Line e and line m are perpendicular. If two lines are perpendicular, they intersect to form right angles. 20. If two angles are supplementary, then the sum of their measures is 180. L1 and L2 are supplementary. Use the Law of Syllogism to make a conclusion. 21. If Kate studies, she will get good grades. If Kate gets good grades, she will graduate. 22. If a, then b. If b, then c. 23. If the weather is wet, the Huskies will not play soccer. If the Huskies do not play soccer, Nathan can stop at the ice cream shop.2-4 Objective&quot;. To connect reasoning in algebra and geometryIn algebra, you use deductive reasoning and properties to solve equations. In geometry, each statement in a deductive argument is justified by a property, definition, or postulate. Some of the properties you need are listed below. Properties of Equality Addition Property Subtraction Property Multiplication Property Division Property Substitution Property Distributive Property If a If a If a If a If a= b, then a + c = b + c.==b, then a - cb, then a . c=b - c. 12.c=b . c. cof=-==band cof=-0, then!!b, then b can replace=ain any expression.a(b + c)ab + ac118Chapter 2Chapter ReviewPropertiesof CongruenceReflexive PropertyAB=AB LA =LASymmetricPropertyIf AB If== CD, then CD == ABLA == LB, then LB == LA. == CD and CD == EF, then AB == EF. LA == LB and LB == LC, then LA == LC.x+3 Q a.~ R 2xTransitivePropertyIf AB IfI£l 24.Algebra Fill in the reason that justifies each step. Given: QS QR x=42s+ RS = QS + 3 + 2x = 42 3x + 3 = 423xb.~ c.~ d.~= 39 x = 13-:«.each statement. 26. Division PropertyUse the given propertyto complete25. Addition Property of Equality If x = 5, then x + 3 = ~. 27. Reflexive Property of EqualityIf2(AX)= 2(BY),of Equality then AX=~.mLY=~29. Transitive Property of Equality Ifx = 5 and 5 = y, then x =~. 31. Distributive 3p - 6q Property28. Symmetric Property If XY = RS, then~. 30. Distributive 2( 4x + 5) Property = 8x + ~of Equality32. Reflexive Propertyof Congruence= 3(~)NM==~2·5 Objective &quot; To prove and apply theorems about anglesA statement that you prove true is a theorem. A proof written as a paragraph is a paragraph proof. You can prove that vertical angles are congruent; that supplements of the same angle are congruent, and that complements of the same angle are congruent. 33. Algebra Find the value of y. 34. For the diagram at the left, find each of the following. a. mLAEC b. mLBED c. mLAEBA(3yB+20t(5y - 16)0E C D35. Complete Given: Prove:the following paragraphproof.L1 L2== L4 == L3 ==~.L1 ~ L4 is .Proof: By the Vertical Angles Theorem, z.I. ==~ and L4 given, so L2 == L3 by the ~ Property of Congruence.Chapter 2Chapter Review119`

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