Geomery Review of PIL KEY Name: Parallel and Inersecing Lines Dae: Per.: PIL01: Use complemenary supplemenary and congruen o compare wo angles. 1) Complee he following definiion: Two angles are Complemenary IFF he sum of heir measures is 90 degrees. 2) Complee he following definiion: Two angles are Supplemenary IFF he sum of heir measures is 180 degrees. 3) In he picure below, name wo pairs of angles ha are Complemenary. 7 & 5 and 3 & 8 4 7 3 5 8 4) In he picure below, name wo pairs of angles ha are Supplemenary. 1 & 2 and 2 & 4 1 m 2=120 2 4 m 4=60 5) Name all angles ha are congruen o A. m A=30 1, 3, 4, & 6 3 m 3=50 1 30 2 150 4 6 3 60 5 7 6) For each scenario below SKETCH an angle maching he descripion. Wrie he angles measuremen as well. a. The complemen of A b. The supplemen of A A 40 140º 50º c. A and B form Linear Pair. d. A and C form Verical Angles. m B 140º m C =40º For each of he following word problems, idenify he angles, wrie an algebraic expression, solve hen answer he quesion. 7) An angle is wice is complemen. Find he angle. x = 2(90-x) x = 180 2x 3x = 180 x = 60 and he complemen = 90-60 = 30 8) Twice an angle is 60 more han is supplemen. Find he wo angles. 2x = 60 + (180 x) 2x = 240 x 3x = 240 x = 80; supplemen = 180-80 = 100. PIL02: wo inersecing lines, idenify Linear Pairs and Verical Angles. In he picure a righ, idenify wo ses of angles ha form 9) How many unique linear pairs do you see in he figure A a righ? (don double coun 1 &2 is he same as 2&1) 12 10) How many unique verical angle pairs do you see in he figure A a he righ? (again, don double coun). 6 11) Is i possible for a pair of angles o be boh verical AND Figure A linear pair? Explain. No, because verical angles are no adjacen angles, and linear pairs are. 12) How many linear pairs in figure B? 24 13) How many verical angle pairs in figure B? 12 14) How many pairs of complemenary angles in figure B? 8 15) How many pairs of supplemenary angles in figure B? 24 Figure B
CAB and BAT are supplemenary CAT is a sraigh angle. Definiion of Supplemenary The measure of wo angles sums o 180 iff he angles are supplemenary. m CAB + m BAT = 180 Proracor Posulae If an angle is a sraigh angle, hen is measure is 180. Angle Addiion Posulae If wo angles are adjacen, hen he sum of heir individual measures equals he measure of he angle formed by heir non-shared sides CAB and BAT form linear pair m CAB + m BAT = m CAT m CAT = 180 Subsiuion Propery of Equaliy If a + b = c and c = d hen a+b = d PIL 03 Linear Pair Theorem Proof If wo angles form a linear pair, B Then he angles are supplemenary. C Pu hese Saemens & in he righ order o form a 2 column proof. Saemens CAB and BAT form linear pair CAT is a sraigh angle. m CAT = 180 A m CAB + m BAT = m CAT m CAB + m BAT = 180 CAB and BAT are supplemenary : CAB & BAT form a linear pair. & CAT is a sraigh angle Prove: CAB and BAT are supplemenary T. Definiion of Sraigh Angle If an angle is a sraigh angle, hen is measure is 180. Angle Addiion Posulae If wo angles are adjacen, hen he sum of heir individual measures equals he measure of he angle formed by heir non-shared sides Subsiuion Propery of Equaliy If a + b = c and c = d hen a+b = d. Definiion of Supplemenary The measure of wo angles sums o 180 iff he angles are supplemenary PIL04 1 and 3 form a linear pair 2 and 3 form a linear pair m 1 + m 3 = 180 m 2 + m 3 = 180 Linear Pair Theorem If wo angles form a linear pair, hen hey are supplemenary. m 1 = 180 - m 3 m 2 = 180 - m 3 m 1 = m 2 Definiion of Linear Pair Two angles form a linear pair iff heir non-shared sides form a sraigh angle 1 and 3 are supplemenary 2 and 3 are supplemenary Subracion Propery of Equaliy If he same number is subraced from boh sides of an equaion, Then he new equaion is equivalen o he original. 1 and 2 are verical angles Definiion of Supplemenary The measure of wo angles sums o 180 iff he angles are supplemenary. Subsiuion Propery of Equaliy Subsiue (m 1) for (180 - m 3) in m 2 = (180 - m 3) Pu hese Saemens & in he righ order o form a 2 column proof. Verical Angles Theorem If wo angles are verical angles Then hey have equal measures, 1 and 2 are verical Angles Prove ha 1 and 2 have equal measures. Saemens 1 and 2 are verical angles 1 and 3 form a linear pair 2 and 3 form a linear pair 1 and 3 are supplemenary 2 and 3 are supplemenary m 1 + m 3 = 180 m 2 + m 3 = 180 m 1 = 180 - m 3 m 2 = 180 - m 3 m 1 = m 2 ( 1 & 2 have he equal measures) 2 1 3 Definiion of Linear Pair Two angles form a linear pair iff heir nonshared sides form a sraigh angle. Linear Pair Theorem (If LP, hen Supplemenary) Definiion of Supplemenary The measure of wo angles sums o 180 iff he angles are supplemenary. Subracion Propery of Equaliy If he same number is subraced from boh sides of an equaion, Then he new equaion is equivalen o he original. Subsiuion Propery of Equaliy Subsiue (m 1) for (180 - m 3) in m 2 = (180 - m 3)
PIL05 a pair of lines cu by a ransversal, idenify he angle pairs by name. Wrie he appropriae name for each pair of angles. Do no use abbreviaions. g) 7 & 5 are Linear Pair h) 4 & 6 are Same Side Inerior Angles i) 3 & 6 are Alernae Inerior Angles j) 2 & 7 are Alernae Exerior Angles k) 1 & 7 are Same Side Exerior Angles l) 1 & 8 are Corresponding Angles g) 1 & 5 are Alernae Exerior Angles 1 2 3 4 8 6 7 5 Name a pair of angles ha saisfy each classificaion. Sae wih respec o which ransversal and wo lines. Example: Corresponding Angles: 5& 3 (c, l and p) Corresponding Angles 4& 8 (, l and m) Verical Angles _ 3& 7 (none,,p and m) Linear Pair 4& 2 (, l and none) _ 2 4 8 5 1 m l Alernae Inerior Angles 5& 7 (c, l and p) Alernae Exerior Angles 2& 9 (, l and p) 3 c 7 9 p Same Side Inerior Angles 1& 5 (c, l and m) Same Side Exerior Angles 2& 8 (, l and m)
PIL06 Prove he four parallel line/ransversal heorems. This pracices only wo of hem! You are responsible for compleing a parially filled in proof (no from scrach). Prove he Alernae Inerior Angles Theorem If wo lines, cu by a ransversal are parallel, Then he alernae inerior angles are congruen. l _ l & m are parallel._ is ransversal Prove 3 6 m 3 6 2 Prove he Same Side Exerior Angle Theorem If wo lines, cu by a ransversal are parallel Then same-side Exerior angles are supplemenary. p & q are parallel._ a is ransversal Prove 1& 3 are supplemenary 3 2 1 a q p Saemens 1) l & m are parallel. is ransversal. 2) 2& 6 are corresponding angles_ 3) _ 2_ _ 6_ 4) 2& 3 are verical angles 4) m 2 = m 3 5) 2 3 5) If Ver.Angles hen 5) 1& 2 form Linear Pair 6) 1& 2 are Supplemenary 6) 3 2 6) Symmeric_prop of 7) m 1 + m 2 = 180 7) 3 6 7) Transiive prop of 8) 1 + 3 = 180 9) 1& 3 are supplemenary 9) Defin of Supplemenary QED QED PIL07 Prove he converse of four parallel line/ransversal heorems. This pracices only wo of hem! The Converse of he Alernae Exerior Angles Theorem If wo lines are cu by a ransversal so ha alernae exerior The Converse of he Same Side Inerior Angles Theorem. If wo lines are cu by a ransversal so ha same side inerior angles are congruen, Then he lines are parallel. a is ransversal 6 4 Prove r & q are parallel Saemens 1) 2) Defin. of Corresp. Angles parallel 3) If ransversal hen corresponding 4) Defin. of Verical Angles 6 5 a 4 q r Saemens 1) p & q are parallel. a is ransversal. 2) 2& 3 are corresponding angles_ 3) 2 3 angles are supplemenary, Then he lines are parallel. is ransversal 3 & 6 are suppl. Prove l & m are parallel Saemens 1) 2) Defin. of corresponding angle parallel 3) If ransversal hen corresponding 4) Defin. of Congruen Angles 5) Defin of Linear Pair 6) If Linear Pair hen Supplem. 7) Definiion of Supplemenary 8) Subsiuion prop of = m l 2 3 6 1) 6 4 1) a is ransversal. 2) 5& 6 are Verical Angles 2) Defin. of Verical Angles 3) 5 6 3) If Verical Angles, hen 4) 5 4 5) 5 & 4 are Corresponding Angles. 6) r & q are parallel 4) Transiive prop of 5) Defin. of Corresponding Angles ransversal 6) If congruen hen parallel corresponding QED 1) 3& 6 are supplemenary 1) is ransversal. 2) m 3 + m 6 = 180 2) Defin. of supplemenary 3) 2& 3 form Linear Pair 3) Defin. of Linear Pair 4) 2& 3 are supplemenary 4) If Linear Pair hen Suppl 5) m 2 + m 3 = 180 5) Defin of supplemenary 6) m 2 + m 3 = m 3 + m 6 6) Subsiuion Prop Equaliy 7) m 3 =m 3 7) Reflexive Prop equaliy 8) m 2 = m 6 8) Subracion prop of = 9) 2 & 6 corresp angles 9) Defin of Corresp. Angles 10) l & m are parallel. ransversal 10) If congruen hen parallel QED corresponding
Which of he following pairs of lines are parallel? Sae he heorem or posulae ha jusifies. If NOT parallel, use indirec reasoning (aka he conraposiive of a heorem we know o be rue) o jusify ha i is NOT parallel. l 20 20 m 160 Parallel: YES or NO? (circle one) Jusificaion: IF_parallel, ransversal and alernae inerior THEN congruen. AND/BUT SINCE al in NOT congruen WE CONCLUDE l and m canno be parallel Parallel: YES or NO? (circle one) Jusificaion: IF ransversal, corresponding, congruen, y 145 35 THEN parallel AND/BUT SINCE corresponding are congruen WE CONCLUDE lines x and y are parallel x 35 w v 108 72 z 72 Which line pair(s) is/are parallel? & z or w and v (circle one or boh pairs) Jusificaion re w&v: IF_ransversal, corresponding, congruen, THEN parallel AND/BUT SINCE linear pair o 108 (a /v) is 72, so corresp are WE CONCLUDE w and v are parallel Jusificaion re &z: IF ransversal, same-side and parallel THEN supplemenary AND/BUT SINCE he angle verical o 72 @ z/w is same side inerior wih 72, hey should be supplemenary. Bu 72 and 72 are no supplemenary. Since hey are NOT supplemenary_ WE CONCLUDE lines and z are no parallel. p q 66 Find he value of he missing variables, if you can. If you canno, sae, Canno Be Deermined from he informaion provide. 3a+27 w x+13 2x + 12 3x - 32 2x - 5 v y - 3 y +10 a = 13 x = 18 ; y = canno be deermined x = 40, y = _95_ If parallel, ransv, AEA, hen congruen. If congruen, hen measures equal 3a + 27 =66 3a = 39 a = 13 If verical angles, hen congruen 2x 5 = x + 13; so x = 18 If parallel, ransversal, sse, hen suppl. Bu, we don know parallel, so we can make an equaion over he river beween w and v. Y canno be deermined. If linear pair, hen supplemenary, if supplemenary, hen add o 180. So 2x + 12 + 3x 32 = 180. 5x = 200; x = 40. If verical angles, hen congruen. 2x + 12 = y 3 2(40) + 12 =y 3; 95 = y.
Perform he consrucions indicaed: Consruc a line perpendicular o he lines shown, hrough he poin A. A A Consruc a line parallel o he line shown by using Corresponding Angles Posulae. Consruc a recangle using AB as one side, and use he lengh of BC for he oher sides.