Algebraic Proof

к E'I Name _________,___...__7 ____ Date _______ Class. ___ ____ lii• Reteach Algebraic Proof А proof is а logical argument that shows а conclusion is true. An algebraic proof uses algebraic properties, including the Distributive Property and the properties of equality. Examples Symbols Properties of Equality Addition lf а= Ь, then а + с= Ь + с. lf х=-4, then х + 4=-4 + 4. Sublraction lf а= Ь, then а - с= Ь - с. lf r+ 1=7, then r+ 1 -1=7 -1. Multiplication lf а= Ь, then ас= Ьс. lf Division lf а=2 and с -=1- О, then -=-. lf 6= 3t, then Reflexive а=а 15=15 Symmetric lf а= Ь, then Ь=а. lf п=2, then 2= п. Transitive lf а= Ь and Ь= с, then а= с. lf у= 32 and 32= 9, then у= 9. Substitution lf а= Ь, then Ь сап Ье substituted for а in any expression. lf х=7, then 2х=2(7). а Ь с с k 2 = 8, then k 2 (2)= 8(2). 6 Зt 3=3. When solving an algebraic equation, justify each step Ьу using а definition, property, or piece of given information. 2(а + 1)=-6 2а + 2=-6 -2 -2 Given equation Distributive Property Sublraction Property of Equality 2а=-8 Simplify. 2а -8 2 2 Division Property of Equality а= -4 Simplify. Solve each equation. Write а justification for each step. Name --------+-k_i_Y_____ Date _______ Class _______ Reteach Algebraic Proof continued When writing algebraic proofs in geometry, you can also use definitions, postulates, properties, and pieces of given information to justify the steps. mLJKM = mLMKL (Sx - 12) 0 = 4х0 х - 12 = О Definition of congruent angles Substitution Property of Equality SuЬtraction Property of Equality Properties of Congruence figure А Symmetric = lf figure А figure А. Examples Symbols figure А = LCDE= LCDE figure В, then figure В= - lf JK со+ DE 6х = 8 + (Зх + 7) 6х = 15 + Зх Зх= 15 s1 с)rм (\ \ tt . r�t . <;tл Ь S: t. ".\ . () О f" 5, m р1 , f'j _ 'щ blr. Pvc D,v х= 5 {,\d 6. mLPQR = mLPQ S + mLSQR 90 ° = 2х0 + (4х-12) 0 LM, then LM = = JK. -----вх·------� Зх 1 7 с 8 о Е PCIE t\ 90 = 6х-12 == lf LN LP and LP LQ, then LN= LQ. = Write а justification for each step. 5. СЕ= L lf figure А= figure В and figure В= figure С, then figure А figure С. Transitive (5х - 12) 0 Addition Property of Equality Х= 12 Reflexive J "9Lr ас( d po s� дdd'. роЕ 102 = 6х 17 =Х = = с;� mM -\,\С Qwp о(< с:19 r. ldentify the property that justifies each statement. 7. lf LABC LDEF, then LDEF 9. lf FG = HJ, then HJ = FG. LABC. 8. L1 = L2 and L2 = �y�r:;,.,•I,� 10. WX::::WX LЗ, so L1 = LЗ. P.o.wn9ruf11c( Name _______________ Date _______ Class _______ Practice А Algebraic Proof For Exercises 1-12, write the letter of each property next to its definition. The letters а, Ь, and с represent real numbers. А. Addition Property of Equality 1. lf а= Ь, then Ь=а. F 2. lf а= Ь, then ас= Ьс. __ G__ = В. Sublraction Property of Equality 3. -АВ -АВ �--=J '--- С. Multiplication Property of Equality 5. lf а= Ь, then а + с= Ь + с. _......1\ __ Е. Reflexive Property of Equality 4. а=а D. Division Property of Equality � 6. а(Ь +с)= аЬ + ас F. Symmetric Property of Equality .J- 7. lf а= Ь and Ь= с, then а= с. _(q�· __ = = 8. lf L� LQ, then /( LQ= LP. ___ 9. lf LA LB and L.8 LC, then LA LC. �-L G. Transitive Property of Equality Н. Substitution Property of Equality = = 1. Distributive Property J. Reflexive Property of Congruence а Ь 1 О. lf а= Ь and с :;t: О, then -=- . с с К. Symmetric Property of Congruence � _.v __ L. Transitive Ргорегtу of Congruence 11. lf а= Ь, then Ь сап Ье substituted for а in any expression. -Н�- 12. lf а= Ь, then а - с= Ь - с. _·в__ 13. Cali measures her textbook and finds that it is 8 inches wide. She wants to know how many centimeters wide her textbook is. The formula to convert inches to centimeters is 2.54i= с, where i is the length in inches and с is the length in centimeters. Fill in the Ыanks to find the answer. The justifications will guide you. 2.54( Given equation 2.54i= с ,2..<,, Q 32- (', )= с =с = 201 Substitution Ргорегtу of Equality Simplify. ? 2. 14. Write а justification for each step. 1----11-----; 7 ОЕ + EF= OF (ix+1) +7=11 !х +8=11 3 1 -х =3 3 Х=9 Symmetric Property of Equality F St1c1 J CttJO pciSt . • - S>1.Ab�f,гиhG1, Ctic S,mpt,�1 � 1А ь fy r-ь Е' > mul t. РС!с i '1 v' f\________ Date ______ Class ______ Name ______ i,j.j Practice В Algebraic Proof Solve each equation. Show all your steps and write а justification for each step. 1 1. - (а+ 1О) = -3 5 t [ tt- IDJ) - - 3(�) 5 1 ( -> а . -10- -i 1о - , i:..->-1 мц lt �ct ,; 1 М...Г'I \ \ � j ,� ° SL( S1м.pt tl� -2� (?,<;"= .. ,.з - рое, bt-Y- 1 2. t +6.5 = 3t - 1.3 -t -t.. - s t,,( bt-v- · Р О f"° s,�pt.f� 2t-/,3 11.'3 � �- �: \ '1 ��� -z._. д tid. PDt $ 1'fY\P li(� 1),v. Рос.S ,м.1JIJ'::;I . sy 3. The formula for the perimeter Р of аrectangle with length .f. and width w is 1 {Ylf\'U. Р = 2(.f. + w). Find the length of the rectangle shown here if the perimeter is 9-½ feet. Solve the equation for .f. and justify each step. р- (,jlJ;_ 2 { l 1 t,J) :: 2- ( f_ + 1 ''4) ')' � е-к. Su ь� 11 • PL�t' 21+ 7-'t� Ь,;+•пЬ ()Ot; .-y-. Pl)c bf �tл 2. , 2 ''2. 2 -i с.7 tt2. - 2. i, :. 2 ! 1 "" 2 _д <;. ,m.,pl' f ':1 __,,,-/ с7112.. -:- 3112 � I е j � 1 V\1\.-pl, �L/ . j_ � ?/i2. J 4. _ ----7х - 3----< 2х + 6 t Зх- 3 J HJ= Н/ + IJ 7х- 3 = (2х+6) + (3х- 3) 7х- 3 = 5х+З 7х=5х+6 2х=6 SPtf. ot:ld. pos t s � ь s t t • PDt I S,mp1,fi_ Ь,11. PD t х=З ldentify the property that justifies each statement. 5. m = п, so п = т. sч m r{W -h1·c 7. KL:LK PD Е 6. LABC = LABC ,.- Re{u., х, v-f ()Dt:. 8. p=qandq=-1,sop=-1. -1 V iл.ft <; rh \1'-(_ Q о+- i1¼tt у т ('{U:. tv'- с., 2. ,�,v РоЕ. � "=i Write а justification for each step. - +n L Pot: СЪ .