60 Reasons 1. L 1 '----- L2 L3 1. Given 2. ME MD · 2018. 8. 29. · 60 15. Statements Key to...

60

15. Statements

Key to Chapter 4, pages 143-145

Reasons

1. L 1 '----- L2 L3 1. Given2. ME M D

AFYE

2. If 2 A of a A areopp. those A are

-------, then the sides

3. EN D G

A are

3. Given4. A ME N AMDG 4. SAS Post.5. L4 L 5 5. Corr. parts of A are

1. A A B C A D E F

2. CB F E ; LB L E3. C X I AB; FY D E4. mL CX B = 90; mLFTE = 90

1. Given2. Corr. parts of3. Given4. Def . of I lines

A are

5. LCXB LFYE 5. Def . of A6. ACXB AFYE 6. A A S Thm.7. CX F Y 7. Corr. parts of A are

16. Given: AABC A D E F ;CX I AB; FY _I_ DE

Prove: CX F Y

17. Given: Isos. AXYZ with XY X Z ;ZA I XY; YB I XZ

Prove: ZA - YB

Statements

1. Z A X Y ; YB X Z2. mL X B Y = 90; mLXAZ = 903. L X B Y - L X A Z4. L X L X5. X Y X Z6. A X B Y A X A Z7. Z A Y B

Statements R e a s o n s

Reasons

1. Given2. Def . of I lines3. Def . of A

4. Ref l. Prop.5. Given6. A A S Thm.7. Corr. parts of A are

Key to Chapter 4, pages 143-145 6 1

Statements R e a s o n s1. CA •--• CB 1. Given2. LCAB L C B A , or

ml_CAB i n Z_ CB A2. Isos. A Thm.

1 13. —mLCAB = —mLCBA2 2 3. Ruh. Prop. of =

4. AX bis. LCAB; BY bis. LCBA. 4. Given15. mL B A X = —mLCAB;2 5. L Bis. Thm.

1mt_ABY = —rn,LCBA2

6. r r t LBAX -= mLABY, or 6. Substitution Prop.LBAX L A B Y

7. AB A B 7. Refl. Prop.8. ABAX A A B Y 8. ASA Post.9. AX B Y 9. Corr. parts of A are

18. Given: Isos. AABC with CA C B ;AX his. LCAB; BY his. LC B A .Prove: AX B Y

62 K e y to Chapter 4, pages 143-145

19. Given: Isos. AXYZ with XY X Z ;M is the midpt. of XY;N is the midpt. of XZ;MP I YZ; NQ I YZ

Prove: MP N Q

1. X Y X Z , or XY X Z2. L I T L Z

1 13.2 XY = 2- X Z4. M is the midpt. of XY;

N is the midpt. of XZ.1 1

5. M Y = --XY; NZ = X Z

6. M Y = NZ7. M P Y Z ; NQ Y Z8. m L M P Y = 90; mLNQZ = 909. L M P Y L N Q Z

10. A M P Y A N Q Z11. M P N Q

1. L A D C / T E D L N C E2. L C D E L D E C L E C D3. mL A DC + mL CDE =

mL FE D + mL DE C =mL NCE + mL E CD

4. m L A D E = mL A DC + mLCDE;mL FE C = mL FE D + mLDEC;mL NCD m L N C E + mL E CD

5. L A D E L F E C L N C D6. A D D E ---: FE E C N C C D7. A A DE A V E C A N C D8. A E F C '--- ND


C 21 . Statements R e a s o n s

1. Given2. Isos. A Thm.

3. Mult . Prop. of =

4. Given

5. Midpt . Thm.

6. Substitut ion Prop.7. Given8. Def . of I lines9. Def . of

10. AAS Thm.

11. Corr. parts of A are

20. Plan for Proof: AFSK is isos., so L F L K . Add LA to both FL and KA, so thatFA = KL and FA K L . Us e the Midpt. Thm. and Substitution Prop. to proveFM K N . Then AFAM A K L N by SAS, and corr. parts AM and LN are

1. Def . of square2. A n equilateral A is also equiangular.3. Add. Prop. of =

4. L Add. Post.

5. Substitut ion Prop.6. Def . of square and equilateral A7. SAS Post.

8. Corr. parts of: ---- A a r e


22. Statements R e a s o n s1. A E F C -'---- ND2. m L A E D = mZ FCE = mL NDC3. m L F E D = mL NCE = mL A DC4. m L A E D + mL FE D =

mL FCE + mL NCE =mL NDC + mL A DC

5. m L A E F = mL A E D + mLFED;mL FCN = mL FCE + mLNCE;mL NDA = mL NDC + mL A DC

6. L A E F L F C N L N D A7. F E N C A D8. A A E F A F C N A N D A9. F A N F A N

10. A F A N is equilateral.

5. Statements R e a s o n s

1. B N A C ; CM L A B2. L B M C and LCNB are rt. A.3. A B MC and ACNB are rt. A.4. M B N C5. B C B C6. A B MC A C N B7. C M B N

Page 148 • CLASSROOM EXERCISES

1. Ex . 21

2. Ex . 21; Corr. parts of A are3. Def . of square4. Add. Prop. of =

5. L Add. Post.

6. Substitut ion Prop.7. Def . of square8. SAS Post.9. Corr. parts of'--- A a r e10. Def . of equilateral A

Page 146 • SELF-TEST 2

1. 2x = 180 - 40; x 7 0 2 . 5x 1 1 = 3x + 3; 2x = 14 = 73. 1 8 0 - (60 + 90) = 304. Since RN I AC and CM I AB, L A NB and LAMC are rt. A and are

AB A C and LA L A , so AABN A A C M by AAS Thm.

1. Given2. Def . of I lines3. Def . of rt. A4. Given

5. Ref l. Prop.6. H L Thm.


1. a . SAS b . Corr. parts of A are c . SAS d . Corr. parts of A are2. a. SAS b . Corr. parts of A are c . A S A d . Corr. parts of A are3. a. SSS b . Corr. parts of A are c . SAS d . Corr. parts of:----- A a r e4. Prove ACPE A G Q E by AAS, so corr. parts CP and GQ are T h e n prove

ACDP A G F Q by HL, so corr. parts L D and L F are

Pages 148-151 • WRI TTEN EXERCISES

A 1 . a. SSS b . Corr. parts of'a, - A a r e - c . S A S d . C o r r . p a r t s o f A a re


2. a. SSS b . Corr. parts of'-- -- A a r e c . A A S d . C o r r. p a r ts o f A a re

3. a . AAS b . Corr. parts of A are c . SAS d . Corr. parts of A are4. a . SAS b . Corr. parts of------ A a r e c . A S A d . C o r r. p a r ts o f A a re

5. a . SAS b . Corr. parts of'- -- A a r e c . H L d . C o r r. p a r ts o f A a re

6. a. I f 2 A of a A are t h e n the sides opp. those A are b . SSSc. Corr. parts of A are d . SAS e . Corr. parts of A are

B 7 . a . 1. L E L A A F K A (SSS) 2 . Z 1 £ 2 (Corr. parts of'----- A a r e

3. A F L J A F K J (SAS) 4 . L J K J (Corr. parts of A areb.Statements R e a s o n s

1. L F 1 0 ' ; LA 7=- K A2. F A F A3. A F L A A F K A4. L i £ 25. F J F J6. A P-7 J AFKJ

7. L J K J

8. a. 1. A P S R A P T R (ASA) 2 . S R T R (Corr. parts of'---- A a r e3. ASRQ A T R Q (SAS Post.) 4 . £ 3 £ 4 (Corr. parts of A are

5. P R bis. LSQT (Def. of A bis.)b. APSR A P T R by ASA, so corr. parts SR and TR are S R and TR are alsocorr. parts of ASRQ and ATRQ, which can now be proved b y SAS. So corr. parts£3 and £4 are a n d PR bis. LSQT.


1. A RS T A X Y Z2. S T Y Z ; L T ----: LZ;

LRST L X Y Z , ormLRST = mL X Y Z1 13. - mL RS T = -m, LXYZ2 2

4. S K bis. LRST; YL bis. L X Y Z.1

5. mL K S T = -2m L R S T ;

1mLLYZ = -2m L X Y Z

6. mL K S T = mLLYZ7. A K S T A L Y Z8. S K Y L

1. Given

2. Ref l. Prop.3. SSS Post.

4. Corr. parts of A are5. Ref l. Prop.6. SAS Post.

7. Corr. parts of ----' A are

1. Given

2. Corr. parts of.•-•-• A a r e3. Mult . Prop. of =

4. Given

5. L Bis. Thm.

6. Substitut ion Prop.7. A S A Post.8. Corr. parts of A are



1.

1. PQ Q R ; PS I SR 1. Given

2.

2. L P Q R and LPSR are rt. A. 2. Def. of I lines

3.

3. A P Q R and APSR are rt. A. 3. Def. of rt. A

4.

4. PQ P S 4. Given

5.

5. P R P R 5. Refl. Prop.

6. m L H D E = 90; mLICFG = 90

6. A P Q R A P S R 6. HL Thin.

L HDE L K F G

7. LQPO L S P O 7. Corr. parts of A are8. PO P O 8. Refl. Prop.

DH F K

9. A ( 2 /30 A SP ()

9. SAS Post.

8. L B L F

10. QO S O 10. Corr. parts of A are

11. 0 is the midpt. of QS. 11. Def. of midpt.


1. DE F G ; GD E P ' 1. Given2. GE G E 2. Refl. Prop.3. AGDE A E F G 3. SSS Post.4. L DE H L F G K 4. Corr. parts of A are5. L HDE and LKEG are rt. A. 5. Given6. m L H D E = 90; mLICFG = 90 6. Def. of rt. A7 L HDE L K F G 7. Def. of8. AHDE A K F G 8. ASA Post.9. DH F K 9. Corr. parts of ------ A are

10. Statements Reasons1. Draw AC and AL 1. Through any 2 pts. there is exactly

one line.

2. CD E D ; Z CDA LEDA 2. Given3. A D A D 3. Refl. Prop.4. A CDA A E D A 4. SAS Post.5. A C -= AE 5. Corr. parts of A are6. A B A F ; BC F E 6. Given7. A A B C A A F E 7. SSS Post.8. L B L F 8. Corr. parts of A are


Statements R e a s o n s1. KL and MN bis. each other at O. 1. Given2. 0 is the midpt. of KL and MN. 2. Def . of his.

BC

3. KO = OL or KO 0 1 / 3. Def . of midpt.

4.

MO = ON or MO O N

5. Z MADLI-- LMBC

4. LKON L L O M 4. Vert . A are5. AKON A L O M 5. SAS Post.6. L K L L 6. Corr. parts of A are7. LKOP L L O Q 7. Vert . A are8. AKOP A L O Q 8. A S A Post.9. PO ----' QO 9. Corr. parts of A are

10. 0 is the midpt. of PQ. 10. Def . of midpt.

1. L M D C L M C D 1. Given2. M D •--• MC 2. If 2 A of a A are t h e n the sides

opp. those A are3. A M M B ; AD BC 3. Given4. A A M D A B M C 4. SSS Post.5. Z MADLI-- LMBC 5. Corr. parts of-

------- A a r e

6. A B A B 6. Refl. Prop.7. A DA B A C B A 7. SAS Post.8. A C B D 8. Corr. parts of A are

13. Given: KL and MN bis. each other at O.

Prove: 0 is the midpt. of PQ.

14. Since DB = EC and BA = CA, then DA = EA. Us ing LA in both A,ADAC:-=•-• A EA B by SAS. So corr. parts AD and LE are and corr. parts

DC and KB are T h e n ADBC A E C B by SAS, so corr. parts LDBC andLECB are Th e r e f o r e , LABC L A C B because they are supps. of

C 15 . Statements Reasons

6. a.

Key to

16.

Chapter 4, page 151

Statements

67

Reasons

1. Z 3 ----' L42. m L 3 + mL 7 1 8 0 ;

mL4 + mL 8 = 180

1. Given2. A Add. Post.

3. L 3 and L7 are supp.; 3. Def . of supp. AL4 and L8 are supp.

4. L 7 -------- L8 4. I f 2 A are supps. of7÷-•-• A , t h e n t h e

2 A are5. AG :--- AF 5. I f 2 A of a A are t h e n the sides

opp. those A are6. L l •-•• L2 6. Given7. A B A G A E A F 7. A S A Post.8. A B A E 8. Corr. parts of A are9. L 5 L 6 9. Given

10. A C A D 10. I f 2 A of a A are t h e n the sidesopp. those A are

11. A B A C A E A D 11. SAS Post.12. B C E D 12. Corr. parts of A are

17. isos.; AX A l ' , AZ AZ, and LXAZ L Y A Z , so AXAZ A Y A Z by SAS. Thenn

7. a.

corr. parts XZ and YZ are a n d AXYZ is isosceles.

Page 151 • MI XED REVI EW EXERCISES

1. Two sides of a A are i f and only if the A opp. those sides are 2 . sometimes3. sometimes 4 . always5. a. •A M

b.

b. K

•b . •A

68 K e y to Chapter 4, page 155

Statements Reasons

median

1. B E C D ; BD CE 1. Given

5. S, T; RS, RT

2. B C '--- BC

a. B E

2. Refl. Prop.

7.

3. A E B C A D C B

Q

3. SSS Post.4. L E B C L D C B 4. Corr. parts of A are5. A B A C 5. I f 2 A of a A are

opp. those A arethen the sides

6. A A B C is isos. 6. Def. of isos. A

Page 155 • CLASSROOM EXERCISES

1. median 2. alt itude 3 . I bis.4. a. SAS b. isos. 5. S, T; RS, RT6. a. B E b. A D c. CP7. a. X Q Y b . SSS Post..

8. a .

10. a .

11.

b.

b. S

1 ?- -L T

b. AF

c. Corr. parts of'----- A a r e d . I f 2 l i ne s f or m ad j. A,

then the lines are 1. e . I bis.

8. a . Yes b . Yes 9 . equilateral10. a . P is equidistant from the sides of LN. b . Q lies on NO.


11. AAOC A B O C by SAS; AAOD A B O D by SAS; AACD A B C D by SSS.

Pages 156-158 • WRI TTEN EXERCISES

A 1 - 5 . Check students' drawings. 1 . b . No 5 . Yes; at the midpt. of the hyp.

B

1. / is the I bis. of BC. 1. Given

1. Given

2. I f a pt. lies on the I his. of a seg.,then it is equidistant from theendpts. of the seg.

3. Trans. Prop.

2. / I BC; Xi s t h e m i d pt . o f B C.

2. Def. of I bis.3. L A X B L A X C 3. I f 2 lines are I , then they form

adj.4. B X C X 4. Def. of midpt.5. A X A X 5. Refl. Prop.6. A A X B A A X C 6. SAS Post.7. A B A C or AB = AC 7. Corr. parts of -= A are


B

6.10.

13.

14.

The 3 L bisectors meet in a pt. 7 . K S , KN 8 . NS , NK 9 . bis. of LSL, A 1 1 . A , F 1 2 . I his. of LF

Statements R e a s o n s1. Pi s on the I bisectors of AB and

BC.

2. P A = PB; PB = PC

3. P A = PC

1. Given

2. I f a pt. lies on the I his. of a seg.,then it is equidistant from theendpts. of the seg.

3. Trans. Prop.


1. Le t X be the midpt. of BC.2. X B X C3. Draw AX.

4. A X A X5. A B = AC or AB A C6. A A X B A A X C7. L l L 28. A X I BC

9. A- is the Ibis. of BC.

10. A is on the I his. of BC.

1. Ruler Post.

2. Def . of midpt.3. Through any 2 pts. there is exactly

one line.

4. Ref l. Prop.5. Given6. SSS Post.7. Corr. parts of A are8. I f 2 lines form a d j . A, t h e n t h elines are9. Def . of I his.

10. A is on AX

69

70

16.



1. B Z bis. LABC. 1. Given2. L P B X L P B Y 2. Def. of L his.

3. P X B A ; PY 1 BC 3. Given

4. mL P X B = 90; mLPYB = 90 4. Def. of I lines5. L P X B L P Y B 5. Def. of6. P B P B 6. Refl. Prop.7. A P X B A P Y B 7. AAS Thin.8. P X P Y or PX = PY 8. Corr. parts of ----' A. are


1. P X B A ; PY I BC 1. Given2. L P X B and LPYB are rt. L. 2. Def. of I lines3. A P X B and APYB are rt. A. 3. Def. of rt. A4. P B P B 4. Refl. Prop.5. P X = PY or PX P Y 5. Given6. A P X B A P Y B 6. HL Thin.

7. L P B X L P B Y 7. Corr. parts of A are8. B P his. LABC. 8. Def. of L his.


1. S and V are equidistant fromE and D.

2. S and V are on the I his. of ED.

3. S V is the I his. of ED.

1. Given

2. I f a pt. is equidistant from theendpts. of a seg., then it is on the Ihis. of the seg.

3. Through any 2 pts. there is exactlyone line.



1. A B A C or AB = AC 1. Given

2. LABC L A C B 2. Isos. A Thm.

3.1 1AB = —2 AC

3. Mult. Prop. of =

4. CM and BN are medians. 4. Given

5. M is the midpt. of AB; 5. Def. of medianN is the midpt. of AC.

6.1 1

MB = -zi.B; N C = y L IC

6. Midpt. Thin.

7. MB = NC or MB N C 7. Substitution Prop.8. BC B C 8. Refl. Prop.9. AMBC A N C B 9. SAS Post.

10. CM B N 10. Corr. parts of A are

22. Given: AB A C ; CM and BN are medians.Prove: CM------ B N

23. Q is on the I his. of PS, so PQ = SQ. S is on the I bis. of QT, so QS = TS. ThenPQ = TS by the Trans. Prop.

24. P is on the bis. of LADE, so the distance from P to BA equals the distance fromP to DE. P is on the bis. of LDEC, so the distance from P to DE equals the distancefrom P to BC. By the Trans. Prop. the distance from P to BA equals the distancefrom P to BC. Thus P is on the bis. of LABC and BP is the bis. of LABC.

25. a. OD is a I b is . of AB, so AD B D . b . OC is a I his. of AB, so AC B C .c. B y parts (a) and (b) above, AD B D and AC B C . Then since CD C D ,ACAD A C B D by SSS, and corr. parts LCAD and LCBD are

C 26 . M N is the I bisector of TS, so MT M S and TN N S . Rt . A MNT r t . AMNSby HL, so corr. parts L TMN and LSMN are A l s o , since TS I RT andTS I MN, RT 11 MN. Then LRTM L T M N (alt. int. L,) a n d A R A S M N(corr. L,) . B y t he S ub st i tu ti on Prop., LRTM AR. Hence, in ARMT, MR MT.

Since MT M S , then MR ------- MS by the Trans. Prop. and TM is a median.


20.

o,',..* ,.A .,...,.....................................................„...„

C.

Reasons

Main St.

Statements

1. A L M N A R S T 1. Given2. LAT L T ; LN R T 2. Corr. parts of A are3. L X and RY are altitudes. 3. Given4. L X X N ; Y T 4. Def . of altitude5. m L X = 90; mZ Y = 90 5. Def . of I lines6. L X = L Y 6. Def . of A7. A L X N A R Y T 7. A A S Thm.8. L X R Y 8. Corr. parts of A are

21. a.

19. a .

Statements

1. A B A C ; BD I AC; CE I AB2. mL B DA = 90; I nLCEA = 903. L B D A L C E A4. L A '---- L A5. A A DB ;'=-1 A A E C

6. B D C E

b.

Reasons

b. The altitudes drawn to the legs of an isos. A are

1. Given2. Def . of I lines3. Def . of A

4. Ref l. Prop.5. A A S Thm.6. Corr. parts of A are

HG H F and JG J E . Also, EH H P andJQ F J . Vert . A are'----, s o A E H F A P H G a n d

AFJ E A Q J G by SAS. Then corr. parts EF and PG Fare a n d EF and GQ are T h e n GQ G P . b . Corr. parts of A are -----', soL HE F L H P G and LJ FE L N G . Then GP 11 E F and GQ E F (I f 2 linesare cut by a trans. and alt. int. A are:- -- - , t h e n t h e l i n e s a r e 1 1 .) . c . P , G , a n d Q a re

( 2 8 . a . A E B C and AB A B . Since AD B D , LDAB L D B A . Us ing alt. in

BE = AC, EO = CO and thus LCEO L E C O . LEOC L A O B (vert. A), so

mLCE0 = —1( 1 8 0 — m L EO C ) a nd m LE BA = —( 180 — mLEOC) . Then

LEAD L A D B and LADB L D B C , so LEAD L D B C . mL E A B =mL E A D + mL DA B and mLCBA = mL CB D + mLDBA, so LEAB L C B A .Then AEAB A C B A by SAS, and corr. parts AC and BE areb. Us ing the A from part (a) above, LEBA L C A B , so OA = OB. Since

2 1

collinear since through a pt. outside a line there is exactly one line 11 t o a given line. A,

LCEO L E B A and EC 11 AB. 229. Since AM is the I bis. of BC, AB A C . L i L 2 so DA bisects LBDF and

AE A F (I f a pt. lies on the bis. of an L, then the pt. is equidistant from the sidesof the L.). Since AE I BD and AF I DF, AAEB and AAFC are rt. A.AAEB A A F C by 111,, so corr. parts BE and CP are

1


27. a. Since EH and FJ are medians,

Page 158 • EXPLORATIONS

Check students' drawings.1. False; true for vertex A of isos. A. 2 . False; true for vertex A of isos. A.3. False; true from rt. L vertex in rt. A.

Page 159 • SELF-TEST 3

1. E A D B ; LAEB L B D A


1. A MP Q A P M N2. M N Q P ; LMPQ L P M N3. M S P R4. A MS N A P R Q

3. a. L J or KJ b . K Z 4 . No

73

1. Given2. Corr. parts of A are3. Given4. SAS Post.


Pages 160-161

1.8.

9.

• CHAPTER REVI EW

AQPR 2 . A TS W 3 . L WYes; SAS

Statements

4. WT 5 . Yes; SSS 6 . No

Reasons

7. Yes; ASA

1. J M L M ; JK LK

2. Def. of a line I to a plane3. m L M K J = 90; I nLMKL = 90

1. Given

2. Ref l. Prop.3. SSS Post.


2. M K 2--= M K

3. A M J K A M L K4. L M J K


1. LJ MK L L M K ; MK I plane P 1. Given2. MK I JK; MK I LK 2. Def. of a line I to a plane3. m L M K J = 90; I nLMKL = 90 3. Def. of I lines4. L MK J L M K L 4. Def. of5. MK M K 5. Refl. Prop.6. AMKJ A M K L 6. ASA Post.7. JK L K 7. Corr. parts of A are

11. ER, EV 1 2 . equilateral 1 3 . 3x = 75; x =- 2514. 3y + 5 -= 25 - y; 4y = 20; y = 5

15. Statements Reasons

1. G H H j ; KJ I HJ 1. Given2. mLGH. I = 90; mLKJ H = 90 2. Def. of .1 lines3. L G HJ L K J H 3. Def. of4. L G L K 4. Given5. H J H J 5. Refl. Prop.6. A G HJ A K J H 6. AAS Thm.

5. I f a pt. lies on the bis. of an angle, then the pt. is equidistant from the sides ofthe angle.

6. I f a pt. is equidistant from the endpts. of a seg., then the pt. lies on the I bis.of the seg.



L i L 2 ; LPQR

1. L l L 2 1. Given

2.

2. WX ------- WY 2. I f 2 A of a A areopp. those A are

then the sides

3. L3 L 4 3. Given

ASA Post.

4. WZ W Z 4. Refl. Prop.

4.

5. AWZX A W Z Y 5. SAS Post.6. ZX Z Y 6. Corr. parts of A are7. AZXY is isos. 7. Def. of isos. A

1. L i L 2 ; LPQR LSRQ 1. Given2. QR Q R 2. Refl. Prop.3. APQR A S R Q 3. ASA Post.4. PR '----- SQ 4. Corr. parts of A are


1. G H H J ; KJ I HJ2. L G HJ and LK J H a r e r t . A .

3. A G HJ and AKJ H are rt. A.4. GJ K H5. H J H J6. A G HJ A K J H7. G H K J

17. 1. A S A 2 . Corr. parts of A are 3 . H L 4 . Corr. parts of A are5. I f 2 lines are cut by a trans. and alt. int. A are'-- --, t h e n t h e l i n e s a r e 1 1

18. a. DG b . F H c . K J

19. I f a pt. lies on the _I_ his. of a seg., then the pt. is equidistant from the endpts. of theseg.

20. I f a pt. is equidistant from the sides of an L, then the pt. lies on the his. of the L.

Pages 162-163 • CHAPTER TEST

1. PT, ADBA 2 . GE, GF; 43 3 . L A , LX; BC, YZ 4 . H L 5 . 1206. median 7 . t he sides of the angle 8 . 7x + 8 = 38 - 3x; 10x = 30; x = 39. Yes; ASA or AAS 1 0 . No 1 1 . Yes; AAS or ASA 1 2 . Yes; EL 1 3 . Yes; SSS

14. Yes; SAS 1 5 . Y, Z 1 6 . W, X 1 7 . AYWX, AZWX, AWYZ, AXYZ18. 8


1. Given2. Def . of I lines3. Def . of rt. A4. Given5. Ref l. Prop.6. H L Thm.

7. Corr. parts of A are -24.


Page 163 • ALGEBRA REVIEW

1. x2 + 5x 6 = 0; (x + 6)(x — 1) = 0; x + 6 = 0 or x — 1 = 0; x = 6, 1

2. n2 — 6n + 8 = 0; (n — 2)(n — 4) = 0; n — 2 = .0 or n — 4 = 0; n = 2,4

3• y2 — 7y — 18 = 0; (y + 2)(y — 9) = 0; y + 2 -= 0 or y — 9 = 0; y = —2, 9

4. x2 + 8x = 0; x(x + 8) = 0; x = 0 or x = —8; x 0, —8

5. Y2 = 13Y; Y2 — 13Y = 0; Y(Y — 13) = 0; y = 0 or y 13 = 0; y 0,13

6. 2z2 + 7z = 0; z(2z + 7) = 0; z = 0 or 2z + 7 = 0; z = 0, —3.5

7. n2 — 144 = 25; n2 169 = 0; (n + 13)(n — 13) = 0; n + 13 = 0 or n — 13 = 0;

n = —13, 138. 50x2 -= 2 00 ; 5 0x2 — 200 = 0; 50(x2 — 4) = 0; (x + 2)(x — 2) = 0; x + 2 = 0 or

x — 2 = 0; x = —2, 29. 50x2 = 2 ; 5 0x2 — 2 = 0; 2(25x2 — 1) = 0; (5x + 1)(5x — 1) = 0; 5x + 1 =- 0 or

5x — 1 = 0; x — 0.2, 0.210. 49z2 = 1 ; 4 9z2 — 1 = 0; (7z + 1)(7z — 1) = 0; 7z + 1 = 0 or 7z 1 = 0;

1 1z_ _7, 7

11. y2 — 6y + 9 = 0; (y — 3)(y 3) = 0; y = 3

12. X2 - 7x + 12 -=- 0; (x — 3)(x 4) -= 0; x — 3 = 0 or x — 4 = 0; x = 3, 4

13. y2 + 8y + 12 = 0; (y + 6)(y + 2) = 0; y + 6 = 0 or y + 2 = 0; y —6, 2

14. t2 + 5t 24 = 0; (t + 8)(t — 3) = 0; t + 8 = 0 or t — 3 = 0; t = —8, 3

15. V2 - 10V + 25 = 0; (v — 5)(v — 5) = 0; v = 5

16. x2 — 3x — 4 = 0; (x + 1)(x — 4) = 0; x + 1 = 0 or x — 4 = 0; x — 1, 4

17. t2 - t - 20 = 0; (t + 4)(t 5) = 0; t + 4 = 0 or t — 5 = 0; t = — 4, 5

18. y2 — 20y + 36 = 0; (y 2)(y — 18) = 0; y — 2 0 or y 18 = 0; y = 2,18

—3 ± + 48 — 3 ± V-577

19. 3 x2 + 3x — 4 = 0; x =

6 6

20. 4y2 — 1 7y + 15 = 0; (4y — 5)(y — 3) = 0; 4y — 5 -= 0 or y — 3 0; y = 1.25, 3

—5 ± V25 — 8 — 5 ±N/F721. X2 + 5x + 2 = 0; x

2 24

22. X2 + 2x — 1 = 0; x —2 ± + 1 ± -2--

25 ± V25 — 12 5 ± VT3

23. X2 - 5x + 3 = 0; x

2 2—3 ± \ /9 + 8 — 3 ± N/17

24. x2 + 3x — 2 = 0; x

2 2


25. y2 - lOy + 25 = 16; y2 - lOy + 9 = 0; (y - 1)(y - 9) = 0; y - 1 = 0 or

y - 9 = 0; y 1 , 926. z2 = 8z 12; z2 - 8z + 12 = 0; (z - 2)(z 6) = 0; z - 2 = 0 or z - 6 = 0;

z = 2, 627. x2 + 5x - 14 = 0; (x + 7)(x - 2) = 0; x + 7 = 0 or x - 2 = 0; x = 7, 2

28. x (x - 50) = 0; x 0 (reject) or x - 50 = 0; x = 5029. X2 - 400 = 0; (x + 20)(x - 20) = 0; x + 20 = 0 or x - 20 = 0;

x = k-20 (reject), x = 2030. X2 - 17x + 72 = 0; (x - 8)(x - 9) = 0; x - 8 = 0 or x - 9 =- 0; x = 8,9

31. 2x2 + x - 3 = 0; (2x + 3)(x - 1) = 0; 2x + 3 = 0 or x - 1 = 0;

3x =2 (reject), x 1

32. 2x2 - 7x 4 = 0; (2x + 1)(x - 4) = 0; 2x + 1 = 0 or x - 4 = 0;

1x -=2 (reject), x = 4

33. 6x2 - 5x 6 = 0; (3x + 2)(2x - 3) = 0; 3x + 2 = 0 or 2x 3 = 0;

x =3 2 (reject), 1 . 5

Page 164 • PREPARING FOR COLLEGE ENTRANCE EXAMS

1. A . (2x + 10) + 3x + (8x - 25) = 180; 13x = 195; x = 15; substituting,2x + 10 = 40, 3x = 45, 8x - 25 = 95, so the triangle is obtuse.

2. C. ( n - 2)180 = 120n; 180n - 360 = 120n; 60n = 360; n = 63. D 4 . C 5 . B 6 . C7. E . 2(2x 3 6 ) + + 2 = 180; 4x 7 2 + x + 2 = 180; 5x = 250; x = 50;

substituting, 2x - 36 = 648. D 9 . B

Page 165 • CUMULATIVE REVIEW: CHAPTERS 1-4

A 1 . Seg. Add. Post. 2 . parallel 3 . obtuse 4 . perpendicular5. 16 6 . I f 3x = 27, then x 9 .7. ( n - 2)180 = 144n; 180n - 360 = 144n; 36n = 360; n -= 108. ( y + 10) + (2y - 31) = 180; 3y = 201; y = 67; substituting, y + 10 = 77,

2y - 31 = 103, 2y - 40 = 94; mLG = 360 - (77 + 103 + 94) = 869. SSS

10. mL 1 = 140 - 65 -= 75; mL3 = mL 2 = 180 - 140 = 40;mZ.4 = 180 - (75 + 40) = 65

11. mL 5 = 90; mZ 7 = 36; mL6 = mL 8 = 180 - (36 + 90) = 5412. Yes; c II d 1 3 . No 1 4 . Yes; a 11 b 1 5 . Yes; a II b


17.

18.

Statements Reasons

1. MO 1 NP; NO -= PO2. LATQO and LPQ0 are rt.3. A NQ O and APO° are rt.4. QO Q O5. A NQ O A P Q O6. L N0 ( 2 L P N

A.A.

1. Given2. Def . of I lines3. Def . of rt. A

4. Ref l. Prop.5. HI , Thm.

6. Corr. parts of A are ----sr.7. Ref l. Prop.8. SAS Post.


7. MO M O8. A MNO A M P O9. M N M P

Since AX is a median, BX

5.

CX• Since AX is an altitude, LAXB and LAXC are

1. M N '--- M P; LNMO LPMO

1.2. MO M O 2.3. A NMO A P M O 3.4. NO P O 4.5. M and 0 lie on the I bis. of NP• 5.

6. MO is the I his. of NP. 6.

B 16 . Statements R e a s o n s

GivenRefl. Prop.SAS Post.Corr. parts of A areI f a pt. is equidistant from theendpts. of a seg., then the pt. lies onthe I his. of the seg.Through any 2 pts. there is exactlyone line.

rt. A. A X A X by the Refl. Prop. So AAXB A A X C by SAS, and corr. partsAB and AC are Th e r e f o r e , AABC is isos.

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60 Reasons 1. L 1 '----- L2 L3 1. Given 2. ME MD · 2018. 8. 29. · 60 15. Statements Key to...

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