4. Statements Reasons
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1 Chpter 9 Answers Prentie-Hll In. Alterntive Ativity 9-. Chek students work.. Opposite sides re prllel. 3. Opposite sides re ongruent. 4. Opposite ngles re ongruent. 5. Digonls iset eh other. 6. Students should summrize their findings from Exerises 3-5. Alterntive Ativity 9-. Chek students work.. Sine C is the midpoint of segment AB it is iseted. Sine point D is rotted CD CE nd segment DE is therefore iseted. 3. Disply the slopes of the sides. 4. Answers my vry. Smple: Sine the slopes of opposite sides re eul the figure is prllelogrm. 5. Chek students work. 6. Chek students work. 7. Mesure slopes. 8. Answers my vry. Smple: Sine the slopes of opposite sides re eul the figure is prllelogrm. Alterntive Ativity 9-4. Chek students work.. Segment CD is onstruted prllel to segment AB. Sine rdii re ongruent in the two irles AC BD. The two rdii re not prllel (y inspetion). 3. Chek students work. 4. Answers my vry. Smple: Bse ngles re ongruent. Digonls re ongruent. 5. AFEB is n isoseles trpezoid. ACEB nd AFDB re prllelogrms. ABCF nd ABED re non-isoseles trpezoids. Retehing 9-. Sttements Resons. Prllelogrm ABCD. Given. AB CD BC DA. Opposite sides of prllelogrm re ongruent. 3. BD DB 3. Reflexive Prop. of 4. ABD CDB 4. SSS 5. A C 5. CPCTC. Sttements Resons. Prllelogrm ACDE;. Given CD BD. C E. Opposite ngles of prllelogrm re. 3. CBD C 3. Isoseles Tringle Theorem 4. CBD E 4. Sustitution 3. Sttements Resons 4. Sttements Resons. Prllelogrm ACDE;. Given CBD E. E C. Opposite ngles of prllelogrm re. 3. CBD C 3. Sustitution 4. CD BD 4. If s of re sides opposite them re. 5. BDC is isoseles. 5. Def. of isoseles tringle Retehing 9-. Sttements Resons. Qudrilterl ABCD. Given AB CD BC DA. AC CA. Reflexive 3. ABC CDA 3. SSS 4. BAC DCA 4. CPCTC DAC BCA 5. AB DC AD BC 5. If lternte interior ngles re then lines re prllel. 6. ABCD is 6. Definition of prlleloprllelogrm. grm. Sttements Resons. Qudrilterl ABCD. Given A C B D. ma mb mc md 360. There re 360 in udrilterl. 3. ma mb 3. Sustitution ma mb ma mb Algeri Simplifition 5. ma mb Division Property of = 6. A nd B re 6. Def. of supplementry supplementry. 7. AD BC 7. If sme-side interior ngles re supplementry lines re prllel. 8. Repet steps 3-7 using D in ple of B to prove AB DC. 9. ABCD is prllelogrm. 9. Def. of prllelogrm. Prllelogrm ACDE;. Given AE BD. AE CD. Opposite sides of prllelogrm re. 3. CD BD 3. Sustitution 4. CBD C 4. Isoseles Tringle Theorem 3. Sttements Resons. BD CD AE BD. Given AE CD. AE CD. Sustitution Geometry Chpter 9 Qudrilterls 4
2 Chpter 9 Answers (ontinued) 3. Sttements Resons 3. ACDE is 3. If one pir of opposite prllelogrm. sides re oth ongruent nd prllel then the udri-lterl is prllelogrm. 4. Sttements Resons. CBD C. Given AE BD AC ED. BD CD. If s of re sides opposite them re. 3. AE CD 3. Sustitution 4. ACDE is 4. If oth pirs of opposite prllelogrm. sides re then the ud. is prllelogrm. Retehing 9-3. m 60; m 30; m3 90. m 80; m 50; m3 50; m m 80; m 00; m3 40; m m 60; m 60; m m 75; m 75; m3 5; m m 45; m 45 Retehing 9-4 midpoint of the se. Therefore the medin hs undefined slope i.e. it is vertil. Sine the se is horizontl segment the medin is perpendiulr to the se. 6. The midpoints re ( 0) ( d e ) ( d e ) nd ( ). Then one pir of opposite sides hs slope of e while the other pir of opposite sides hs slope of d. Therefore the figure is prllelogrm sine opposite sides re prllel. Prtie 9- Exmple Exerises. Given. Def. of. Def. of d. Alternte Interior Angle Thm. e. Alternte Interior Angle Thm. f. Reflexive Prop. of g. ASA Post. h. CPCTC. PLUM is. Given PME ULE lines form lt. int. s. MP UL Def. of MP LU MPE LUE Opp. sides of re. lines form lt. int. s. MPE LUE ME LE ASA CPCTC ( ) Retehing 9-5. B(x k m). Z( 0) ; W(0 ) 3. S( ) ; T(0 ) 4. Eh side hs length nd so it is rhomus. One pir of opposite sides hve slopes of the other pir hve slopes of. Therefore sine ()() the rhomus hs four right ngles nd is sure. 5. Eh side hs length of. Therefore it is rhomus. 6. C(x k m) Retehing 9-6. Eh digonl hs length ( ).. The midpoints re ( ) nd ( ). The line onneting the midpoints hs slope of 0 nd is therefore prllel to the third side. 3. The midpoints re ( 0) ( ) ( ) nd (0 ). The segments joining the midpoints eh hve length. 4. The midpoints re ( ) ( ) ( ) nd ( ). The udrilterl formed y these points hs sides with slopes of 0 0 undefined nd undefined. Therefore the sides re vertil nd horizontl nd onseutive sides re perpendiulr. 5. The medin meets the se t (0 0) the 4 PE UE CPCTC PU nd LM iset eh other t E. Def. of iset Prtie 9- Mixed Exerises ; 40; ; 0; ; 45; 7. 5; 5; ; 05; ; 7; 08; ; 98; Prtie 9- Exmple Exerises. The def. of segment isetor; HJG; HJI; SAS Postulte; CPCTC; ; ; the definition of. No; the figure ould e kite. 3. Yes; opposite sides re y the onverse of the Alternte Interior Angle Thm. 4. Yes; pir of ongruent sides. 5. Yes; lternte interior ngles re ongruent y CPCTC so oposite sides re y the onverse of the Alternte Interior Angle Thm. 6. No; the ongruent opposite sides do not hve to e. 7. Yes; the ongruent opposite sides re lso y the Alternte Interior Angle Thm. 8. Yes; oth pirs of Qudrilterls Geometry Chpter 9 Prentie-Hll In.
3 Chpter 9 Answers (ontinued) Prentie-Hll In. opposite sides re ongruent. 9. Yes; oth pirs of opposite ngles re ongruent. 0. yes. no. no 3. yes Prtie 9- Mixed Exerises. no. yes 3. yes 4. no 5. yes 6. yes 7. x ; y 3 8. x 6; y 3 9. x 64; y 0 0. x 8; the figure is euse oth pirs of opposite sides re ongruent.. x 40; the figure is not euse one pir of opposite ngles is not ongruent.. x 5; the figure is euse the ongruent opposite sides re y the onverse of the Alternte Interior Angle Thm. 3. Yes; the digonls iset eh other. 4. No; the ongruent opposite sides do not hve to e. 5. No; the figure ould e trpezoid. 6. Yes; oth pirs of opposite sides re ongruent. 7. Yes; oth pirs of opposite sides re y the onverse of the Alternte Interior Angle Thm. 8. No; only one pir of opposite ngles is ongruent. 9. Yes; one pir of opposite side is oth ongruent nd. 0. No; only one pir of opposite sides is ongruent. Prtie 9-3 Exmple Exerises. 5; 5; 5. 50; 90; 90; ; 68; 68; ; 90; 60; ; 8; m 6. 3; 90; 58; in ; 90; 90; m 8. 57; 57; m 9. 50; 50; 90; m 0. 6; 90; m. retngle. x 85; y rhomus. x 70; y 0 3. retngle 3. x 8; y 8 4. retngle 4. x 46 y rhomus 5. x 55; y rhomus retngle 7. x 50; y rhomus 8. 3 m Prtie 9-3 Mixed Exerises. rhomus. 7; 54; 54; 7. retngle. 7; 36; 8; retngle 3. 37; 53; 06; rhomus 4. 59; 90; 90; retngle 5. 60; 30; 60; rhomus 6. ; 68; 68; Def. of rhomus 7. RQT 7. Def. of ngle isetor 7d. Reflexive Prop. of 7e. TQU 7f. CPCTC 7g. Angle Addition Postulte 7h. mqur 7i. Distriutive Prop. 7j. Def. of right ngle 7k. Sustitution 7l. Def. of 8. 90; 90; 9; m 9. 70; 90; 70; in ; 90; 90; m Prtie 9-4 Exmple Exerises. 76.5; ; ; ; ; ; ; ; 7 9. Given 9. Bse ngles of n isos. trpezoid re. 9. QT 9d. SAS 9e. QS RT Prtie 9-4 Mixed Exerises ; ; ; ; 9. 96; ; ; ; x 6; y 6 8. x x 5; y 5. x 4; y 5 Prtie 9-5 Exmple Exerises. (d ). ( 0) 3. ( 0) 4. ( 0) 5. C(0) ; K( 0) 6. E(0 ) ; F( ) 7. (5 4) 8. (9 ) 9. ( ) ( ) Prtie 9-5 Mixed Exerises. (.5 );. (.5 ); 4 3. (0.5 0); 4. (0.5 ); E( 3) ; I(4 0) 4. O(3 ) ; M(3 ) ; E(3 ) 5. D(4 ) ; I(3 0) 6. T(0 ) ; A( 4) ; L( ) 7. (4 ) 8. ( 0) Prtie 9-6 Exmple Exerises. (6 0). (5 ). d. e. ( ). (0 ). ( 0). ( ) d. e. ( ) f. 3. (8 0) 3. (9 8) d. (8 4) 3e. (9 0) 3f. (0 4) ( 0) d. ( 3) 4e. 3 (4 6) 4f. ( 3) Prtie 9-6 Mixed Exerises p.. ; ; p y mx p ; y p (p). d. y p p x ; y rp x r p (r p) p p rp p ; ; intersetion t (r e. r p rp y f. (r ) g. ; ; ; y r h. ; y ; y rp rp r y r r x r r ) y mx r (r) (r p) r rp ; intersetion t (r p i. (r p rp ) ). ( 0). ( ). (.5 0.5) d. 3. (4 0) 3. ( 3) d. ( ) 3e. 4. The oordintes for D re (0 ). The oordintes for C re ( 0). Using these oordintes the lengths of DC nd HP n e determined: DC ( 0) (0 ) 4 4 HP (0 ) (0 ) 4 4 DC HP so DC HP Geometry Chpter 9 Qudrilterls 43
4 Chpter 9 Answers (ontinued) Chekpoint. 78; 06; ; 00; ; 67; x 4; y 4 5. x 4; y 6. x 3; y Answers my vry. Smple: If udrilterl hs one pir of opposite sides tht re oth prllel nd ongruent then the udrilterl is prllelogrm. Chekpoint.. x 63; y x 6; y 6 5. Answers my vry. Smple: y (0 ) ( 0) ( 0) (0 ) 6. In ll rhomuses eh digonl isets the other digonl nd pir of ngles. In ll kites only one digonl isets the other digonl nd pir of ngles. Chpter Assessment Form A. 5 m. 3 in m 4. Answers my vry. Smple: x Chpter Assessment Form B. 8 in.. 6 m 3. m 4. All prllelogrms tht re not rhomuses (or sures suset of rhomuses) re not divided into four ongruent tringles y their digonls. For exmple given prllelogrm ABCD (tht is not rhomus or sure) whose digonls interset t E AB BC ABE nd CBE shre segment BE nd AE CE euse digonls of prllelogrm iset eh other. However AB BC so not ll the sides re nd ABE CBE. 5. x 9; y x 5; y x ;. A. N(0 ); ; ( y 6 X( 0) ) 3. N(0 ); X( 0); ( ) 4. N(0 0); X( ); ( ) 5. 50; ; ; ; 90; 76; 8 m 9. 8; 90; 7; 308 in. 0. 4; 90; 48; 0 in.. The midpoints of segments FO OR RD nd FD re: S( ) A( ) T( 0) E(0 ). The lengths of segments SA AT TE nd SE re: SA AT TE SE. By the definition of rhomus SATE is rhomus.. Answers my vry. Smple: 3. no 4. yes 5. yes 6. no x 33; y x 30; y D. ; ; ( D( 0) E( ) ) 3. ; ; ( ) 4. ; ; ( D( 0) E(0 ) D(0 ) E( 0) ) 5. 8; ; ; ; 90; 48 m 9. ; 68; 5 in ; 37; 96 m. The lengths of segments AB BC nd AC re: AB j k BC k l AC l j. Thus the perimeter of ABC is l j j k k l. The midpoints of segments nd re: M( ) N( ) O( l j AB BC AC k l j k 0). The length of segments nd re: MN MN NO MO NO j k MO k l (l j). Thus the perimeter of MNO is whih is (l j j k k l ) the perimeter of ABC.. All rhomuses (nd sures suset of rhomuses) re divided into four ongruent tringles y their digonls. For exmple given rhomus ABCD whose digonls interset t E AB BC CD DA y definition of rhomus. Then AE CE nd BE DE euse the digonls of rhomus iset eh other. Thus y SSS ABE CBE CDE ADE. 3. no 4. yes 5. yes 6. yes Alterntive Assessment Tsk. Answers my vry. Smple: AB CD BC AD AB CD BC AD ABD BDC ACD BAC CBD BDA CAD BCA BE ED AE EC ABC CDA BCD BAD. C E F Soring Guide 3 Student lists ll sttements urtely in prt nd gives orret nswer in prt. Student gives mostly orret nswers ut with some errors. Student gives nswers tht fil to demonstrte understnding of properties of prllelogrms. 0 Student mkes little or no effort. Prentie-Hll In. 44 Qudrilterls Geometry Chpter 9
5 Chpter 9 Answers (ontinued) Tsk. Answers my vry. Smple: Q R T P S Angles re s shown. Let the digonls interset t point T. Then: QT TS PT TR 3 PR 3 nd QS.. Prllelogrms Soring Guide 3 Student gives orret oordintes nd vlid proof. Student gives nswers or proof tht ontin minor errors. Student gives inorret oordintes in prt or poorly onstruted proof in prt. 0 Student mkes little or no effort. Cumultive Review. B. D 3. D 4. C 5. A 6. B 7. C 8. D 9. C 0. D. B. D 3. C 4. B 5. B 6. Answers my vry. Smple: B C Prentie-Hll In. Retngles Sures Rhomuses Soring Guide: 3 Student gives urte nd omplete nswers nd digrm. Student gives nswers nd digrm tht re mostly urte. Student gives nswers or digrm ontining signifint errors. 0 Student mkes little or no effort. Tsk 3 x 90 (Digonls of kite re.) y 5 (Def. of isos. trpezoid) z 75 (Bse ngles of isos. trp. re.) Soring Guide 3 Student gives orret nswers nd resons. Student gives mostly orret nswers nd resons. Student gives inorret nswers nd resons. 0 Student mkes little or no effort. Tsk 4. Q (5 5) ; S (5 5 ). Slope of PR 5 0. Slope of QS 5 (5 ). Sine the produt of their slopes PR QS. A D Sine ABC nd CDA shre side AC we n prove tht the two tringles re ongruent y SSS euse AB CD nd BC AD. 7. Answers my vry. Smple: with line symmetry without line symmetry 8. Answers my vry. Smple: In Euliden Geometry two lines interset in one point. In Spheril Geometry two lines interset in two distint points. Stndrdized Test Preprtion. A. B 3. D 4. B 5. D 6. A 7. C 8. B 9. B 0. C. C. A 3. D 4. C 5. C 6. C 7. A 8. D 9. Answers my vry. Smple:. ABCD is sure. (Given). AB BC CD AD (Def. of sure) 3. ma mb mc md 90 (Def. of sure) 4. ABC BCD (SAS) 5. AC BD (CPCTC) 0. Answers my vry. Smple: If you know the mesure of one ngle in trpezoid the djent ngle on the sme leg ut opposite se is sme-side interior ngle nd is therefore supplementry to the first ngle. The ngle on the opposite leg ut the sme se djent to the first ngle is ongruent to the first ngle. The mesure of the lst ngle n e determined y sutrting the sum of the mesures of the three known ngles from 360. Geometry Chpter 9 Qudrilterls 45
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