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Class i fy ing Quadr i l a te ra l sClass i fy ing Quadr i l a te ra l s
Pages 290-293 Exercises
1. , rectangle,rhombus, square
2. parallelogram
3. trapezoid
4. , rhombus
5. kite
6. trapezoid, isosc.trapezoid
7. rhombus
8. parallelogram
9. rhombus
10. rectangle
11. kite
12. isosc. trapezoid
13. rhombus
14. kite
GEOMETRY LESSON 6-1GEOMETRY LESSON 6-1
Class i fy ing Quadr i l a te ra l sClass i fy ing Quadr i l a te ra l s
15. trapezoid
16. rectangle
17. quadrilateral
18. isos. trapezoid
19. x = 11, y = 29; 13,13, 23, 23
20. x = 4, y = 4.8; 4.5,4.5, 6.8, 6.8
21. x = 2, y = 6; 2, 7, 7, 2
22. x = 1; 4, 4, 4, 9
23. x = 3, y =5; 15, 15,15, 15
24. x = 5, y = 4; 3, 3, 3, 3
25. 40, 40, 140, 140; 11,11, 15, 32
GEOMETRY LESSON 6-1GEOMETRY LESSON 6-1
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Class i fy ing Quadr i l a te ra l sClass i fy ing Quadr i l a te ra l s
26. 58, 58, 122, 122; 6,6, 6, 6
27. rectangle, square,trapezoid
28. Check studentswork
29-34. Answers mayvary. Samples aregiven.
29.
30.
31. Impossible; atrapezoid with one rt.
must haveanother, since twosides are .
32.
33.
GEOMETRY LESSON 6-1GEOMETRY LESSON 6-1
Class i fy ing Quadr i l a te ra l sClass i fy ing Quadr i l a te ra l s
34.
35. A rhombus has 4sides, while a kitehas 2 pairs of adj.sides , but no opp.sides are . Opp.sides of a rhombus
35. (continued)are , while opp. sides of a kite are not .
36.
37. True; a square is both a rectangle and a rhombus.
38. False; a trapezoid only has one pair of sides.
GEOMETRY LESSON 6-1GEOMETRY LESSON 6-1
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Class i fy ing Quadr i l a te ra l sClass i fy ing Quadr i l a te ra l s
39. False; a kite doesnot have opp.sides.
40. True; all squaresare .
41. False; kites are not.
42. False; onlyrhombuses with rt.
are squares.
43. Rhombuses; all 4sides arebecause they come
43. (continued)from the same cut.
44-45. Check students
work.
46. some isos.trapezoids, sometrapezoids.
47. , rhombus,rectangle, square
48. rectangle, square
49. rhombus, square, kite,some trapezoids
50. A trapezoid has onlyone pair of sides.
51-54. Check students
sketches.
51. rectangle, , kite
52. rhombus,
53. square, rhombus,
54. rhombus, , kite
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GEOMETRY LESSON 6-1GEOMETRY LESSON 6-1
Class i fy ing Quadr i l a te ra l sClass i fy ing Quadr i l a te ra l s
55. Answers may vary.Sample:a.
N can be
anywhere on thecan be anywhereon the y- axisexcept (0, 0),(0, 2), and(0, 2).
b. For points N mentioned
55. b. (continued)above, KL = LM and KN = NM , butKL KN .
56-59. Explanationsmay vary. Samplesare given.
56. , rectangle,trapezoid
57. , kite, rhombus,trapezoid, isos.Trapezoid.
58. kite, , rhombus,trapezoid, isos.trapezoid
59. , rectangle,square, rhombus,kite, trapezoid
60. C
61. I
62. C
63. H
=/
GEOMETRY LESSON 6-1GEOMETRY LESSON 6-1
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Class i fy ing Quadr i l a te ra l sClass i fy ing Quadr i l a te ra l s
64. [2] Slope of AB is
. The slope
of BC is 1, so AB and BC are not
. Since oneis not a rightand a rectanglerequires all 4to be right ,the figure couldnot be arectangle.
[1] incorrect slopeOR failure torecognize theinformation
64. [1] (continued)provided by theslopes.
65. Yes; the sum of thelengths of any 2sides is greater thanthe third side.
66. No; 5 + 7 20
67. No; 3 + 5 8
68. 28 mm
69. 16 mm
70. 12 mm
71. 82
72. 90
73. 58
74. y = 3 x + 4
32
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>
GEOMETRY LESSON 6-1GEOMETRY LESSON 6-1
Proper t i es o f Para l l e logramsProper t i es o f Para l l e lograms
Pages 297-301 Exercises
1. 127
2. 67
3. 76
4. 124
5. 100
6. 118
7.
8. 4
9. 4
10. 3; 10, 20, 20
11. 22; 18.5, 23.6, 23.6
12. 20
13. 18
14. 17
15. 12;m Q = m S = 36,m P = m R = 144
16. 6;m H = m J = 30,m I = m K = 150
17. x = 6, y = 8
18. x = 5, y = 7
19. x = 7, y = 10
20. x = 6, y = 9
21. x = 3, y = 4
22. 12; 24
34
GEOMETRY LESSON 6-2GEOMETRY LESSON 6-2
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Proper t i es o f Para l l e logramsProper t i es o f Para l l e lograms
23. Pick 4 equallyspaced lines on thepaper. Place thepaper so that the
first button is on thefirst line and the lastbutton is on thefourth line. Draw aline between thefirst and lastbuttons. Theremaining buttonsshould be placedwhere the drawnline crosses the 2lines on the paper.
24. 3
25. 3
26. 6
27. 6
28. 9
29. 2.25
30. 2.25
31. 4.5
32. 4.5
33. 6.75
34. BC = AD = 14.5 in.;AB = CD = 9.5 in.
35. BC = AD = 33 cm;AB = CD = 13 cm
36. a. DC
b. AD
c.
d. Reflexive
e. ASA
f. CPCTC
GEOMETRY LESSON 6-2GEOMETRY LESSON 6-2
Proper t i es o f Para l l e logramsProper t i es o f Para l l e lograms
37. a. Given
b. Def. of
c. If 2 lines are ,then alt. int. are
.
d. If 2 lines are ,then alt. int. are
.
e. Reflexive Prop. of .
37. (continued)f. ASA h. CPCTC
g. ASA i. CPCTC
38.
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Proper t i es o f Para l l e logramsProper t i es o f Para l l e lograms
39. 38, 32, 110
40. 81, 28, 71
41. 95, 37, 37
42. The lines goingacross may not besince they are notmarked as .
43. 18, 162
44. 60
45. x = 15, y = 45
46. x = 109, y = 88,z = 76
47. x = 25, y = 115
48. x = y = 6
49. x = 10, y = 4
50. x = 12, y = 4
51. x = 0, y = 5
52. x = 9, y = 6
53. The opp. are ,so they have =
53. (continued)measures.Consecutive aresuppl., so their sum
is 180.
54. a. Answers mayvary. Checkstudents work.
b. No; the corr. sidescan be but the
may not be.
55. a. Given
b. Def of as
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GEOMETRY LESSON 6-2GEOMETRY LESSON 6-2
Proper t i es o f Para l l e logramsProper t i es o f Para l l e logramsGEOMETRY LESSON 6-2GEOMETRY LESSON 6-2
56. Answers may vary. Sample:1. LENS and NGTH are s.
(Given)
2. ELS ENS and GTH GNH (Opp. of a are .)
3. ENS GNH (Vertical are .)
4. ELS GTH (Trans. Prop. of )
57. Answers may vary. Sample: In LENS andNGTH , GT EH and EH LS by the def. of a .Therefore LS GT because if 2 lines are to thesame line then they are to each other.
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55. (continued)
c. Opp. Sides of aare .
d. Trans. Prop. of
e. If 2 lines are tothe same line,then they are
to each other.f. If 2 lines are ,
then the corr.are .
g. Trans. Prop. of
h. AAS
i. CPCTC
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Proper t i es o f Para l l e logramsProper t i es o f Para l l e lograms
58. Answers may vary. Sample:1. LENS and NGTH are . (Given)
2. GTH GNH (Opp. of a are .)
3. ENS GNH (Vertical are .)
4. LEN is supp. to ENS (Cons. in a are suppl.)
5. ENS GTH (Trans. Prop. of )
6. E is suppl. to T. (Suppl. of are suppl.)
59. Answers may vary. Sample: In RSTW and XYTZ , R T andX T because opp. of a are . Then R X by the Trans.
Prop. of .
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GEOMETRY LESSON 6-2GEOMETRY LESSON 6-2
Proper t i es o f Para l l e logramsProper t i es o f Para l l e lograms
60. In RSTW and XYTZ , XY TW and R S TW by the def. of a
. Then XY RS because if 2lines are to the same line, thenthey are to each other.
61. AB DC and AD BC by def. of .2 3 and 1 4 because if
2 lines are , then alt. int. are .3 4 because if 2 are eachto 2 , then they are . By
Def. of bisect, AC bisects DCB .
62. a. Given: 2 sides and theincluded of ABCD areto the corr. parts of WXYZ .Let A W , AB WX and
62. a. (continued)AD WZ . Since opp. of aare , A C and W
Y . Thus C Y by theTrans. Prop. of . Similarly,opp. sides of a are , thusAB CD and WX ZY . Usingthe Trans. Prop. of , CD ZY . The same can be done toprove BC XY . Since consec.
of a are suppl., A issuppl. to D , and W issuppl. to Z . Suppls. of are , thus D Z . Thesame can be done to prove
B X . Therefore, since allcorr. and sides are ,
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GEOMETRY LESSON 6-2GEOMETRY LESSON 6-2
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Proper t i es o f Para l l e logramsProper t i es o f Para l l e lograms
62. a. (continued)ABCD WXYZ.
b. No; opp. and sides are not
necessarily in a trapezoid.
63. 10
64. 11
65. 126
66. 126
67. 160
68. 148
69. 42
70. rhombus
71. parallelogram
72. AC DB
73. 49
74. 131
75. 49
76. 131
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GEOMETRY LESSON 6-2GEOMETRY LESSON 6-2
Proving tha t a Quadr i l a te ra l I s a Para l l e logramProving tha t a Quadr i l a te ra l I s a Para l l e logram
Pages 307-310 Exercises
1. 5
2. x = 3, y = 4
3. x = 1.6, y =1
4.
5. 5
6. 13
7. Yes; both pairs of opp. sides are .
8. No; the quad. Couldbe kite.
9. Yes; both pairs of opp. are .
10. No; the quad. couldbe a trapezoid.
11. Yes; both pairs of opp. sides arebecause alt. int.are .
12. Yes; one pair of oppsides are and .
13. Yes; both pairs of opp. sides are .
14. No; opp. sides arenot .
15. No; the quad. couldbe a kite.
16. It remains abecause the shelvesand connectingpieces remain .
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GEOMETRY LESSON 6-3GEOMETRY LESSON 6-3
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Proving tha t a Quadr i l a te ra l I s a Para l l e logramProving tha t a Quadr i l a te ra l I s a Para l l e logram
17. a. bisect
b. XR
c. XYR
d. ASA
e. alt. interior
18. Opp. sides of aquad. are if andonly if the quad. is a
19. a. Distr. Prop.
b. Div. Prop. of Eq.
c. AD BC , AB DC
d. If same-side int.are suppl., thelines are .
e. Def. of
20. Yes; both pairs of opp. are .
21. No; the figure couldbe a kite.
22. Yes; a pair of opp.sides is and .
23. No; the figure could
be a trapezoid.
24. Yes; the both pairsof opp. sides are
.
25. Yes; diag. bisecteach other.
26. x = 15, y = 25
27. x = 3, y = 11
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GEOMETRY LESSON 6-3GEOMETRY LESSON 6-3
Proving tha t a Quadr i l a te ra l I s a Para l l e logramProving tha t a Quadr i l a te ra l I s a Para l l e logram
28. c = 8, a = 24
29. k = 9, m = 23.4
30. Answers may vary.Sample:
31.
32. (4, 0)
33. (6, 6)
34. (2, 4)
35. You can show a quad. is a if both opp. sidesare , if both opp. are , if opp. sides are ,if diag. bisect each other, if all consecutive aresuppl., if one pair of opp. sides are both and .
36. Answers vary. Sample:1. TRS RTW (Given)
2. RS TW , SRT WTR (CPCTC)
3. SR WT (If alt. int. are , then lines are .)
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Proving tha t a Quadr i l a te ra l I s a Para l l e logramProving tha t a Quadr i l a te ra l I s a Para l l e logram
36. (continued)4. RSTW is a . (If one pair of opp. sides are
and , then it is a .)
37. Answers may vary. Sample:1. AB CD , AC BD (Given)
2. ACDB is a . (If opp. sides are , then itis a .)
3. M is the midpoint of BC . (The diag. of abisect each other.)
4. AM is a median. (Def. of a median)
38. G (4, 1), H (1, 3)
39. C
40. F
41. C
42. H
GEOMETRY LESSON 6-3GEOMETRY LESSON 6-3
Proving tha t a Quadr i l a te ra l I s a Para l l e logramProving tha t a Quadr i l a te ra l I s a Para l l e logramGEOMETRY LESSON 6-3GEOMETRY LESSON 6-3
43. [2] Statements Reasons
1. NRJ CPT (Given)
2. NJ CT (CPCTC)
3. NJ TC (Given)
4. JNTC is a . (If opp. sides of aquad. are bothand , thenthe quad. is a .)
[1] proof missing steps
44. [4] a. 6x = 7 x 11;x = 11
b. Yes; m ABC = m CDE = 66
c. Yes; BD FE and BF DE
[3] one or more error in calculating x
[2] one explanationis incorrect
[1] only part (a)answered
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Proving tha t a Quadr i l a te ra l I s a Para l l e logramProving tha t a Quadr i l a te ra l I s a Para l l e logramGEOMETRY LESSON 6-3GEOMETRY LESSON 6-3
45. a = 8, h = 30, k = 20
46. m = 9.5, x = 15
47. e = 17, f = 11, c = 204
48. It is given that AD BC andDAB CBA . By the Reflexive
Prop. of AB AB , thus DAB CBA by SAS, so AC BD by
CPCTC.
49. If a quad. is a , then the diag.bisect each other; if the diag. of aquad. bisect each other, then it is a
.
50. If two lines and a transversal formcorr. , then the two lines are ; if two lines are , then a transversalforms corr. .
51. If the prod. of the slopes of twononvertical lines is 1, then theyare ; if two nonvertical lines are
, then the prod. of their slopes os 1.
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Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
Pages 315-318 Exercises
1. 38, 38, 38, 38
2. 26, 128, 128
3. 118, 31, 31
4. 33.5, 33.5, 113,33.5
5. 32, 90, 58, 32
6. 90, 60, 60, 30
7. 55, 35, 55, 90
8. 60, 90, 30
9. 90, 55, 90
10. 4; LN = MP = 4
11. 3; LN = MP = 7
12. 1; LN = MP = 4
13. 9; LN = MP = 67
14. ; LN = MP = = 9
15. ; LN = MP = 12
16. Impossible; if thediag. of a are ,
16. (continued)then it would have tobe a rectangle andhave right .
17. Yes; diag. in amean it can be arectangle with 2 opp.sides 2 cm long.
18. Impossible; in a ,consecutive mustbe supp., so allmust be right .This would make it arectangle.
53
293
23
52
12
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Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
19. Impossible; if thefigure is a , thenthe opp. thebisected is also
bisected, and thefigure is a rhombus.But the sides are not
.
20. Yes; the arebisected so it couldbe a rhombus whichis a .
21. Yes; the diag. areso it could be asquare which is a .
22. The pairs of opp.sides of the frameremain , so theframe remains a .
23. After measuring thesides, she canmeasure the diag. If the diag. are , thenthe figure is arectangle by Thm. 6-14.
24. Square; a square isboth a rectangle anda rhombus, so itsdiag. have the same
24. (continued)properties of both.
25-34. Symbols may
vary. Samples aregiven:parallelogram:rhombus:rectangle:square:
25. ,
26. , , ,
27. , , ,
s
R
S
R
S
R S
R S
Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
28. , , ,
29. ,
30. , , ,
31. , , ,
32. ,
33. ,
34. ,
35.
Diag. are , diag.are .
36.
Diag. are and .
37.
Diag. are , diag.are .
38. a. Opp. sides areand ; diag. bis.each other; opp.are ; consec.are suppl.
b. All sides are ;diag. are .
c. All are rt. ;diag. are bis. of each other; eachdiag. bis two .
R S
S
R S
R S
S
R S
R S
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Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
39-44. Answers mayvary. Samples aregiven.
39. Draw diag. 1, andconstruct its midpt.Draw a line throughthe mdpt. Constructsegments of lengthdiag. 2 in opp.directions from mdpt.Then, bisect thesesegments. Connectthese mdpts. with the
endpts. of diag. 1 .
40. Construct a rt. ,and draw diag. 1from its vertex.Construct the from
the opp. end of diag.1 to a side of the rt.. Repeat to other
side.
41. Same as 39, butconstruct a line atthe midpt. of diag. 1.
42. Same as 41 exceptmake the diag. =.
43. Draw diag. 1.
43. (continued)Construct a at a pt.different than themdpt. Construct
segments on theline of length diag. 2in opp. directionsfrom the pt. Then,bisect thesesegments. Connectthese midpts. to theendpts. of diag. 1.
44. Draw an acute withthe smaller diag. asa side. Construct theline to the other
Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
44. (continued)side through the non-vertex endpt. of thesmaller diag. Drawan arc with compassset to the length of the larger diag. fromthe non-diag. side of the , passingthrough the line.Draw the larger diag.,and then draw thenon- sides of thetrapezoid.
45. Yes; since all rightare , the opp. are
45. (continued)and it is a .
Since it has all right, it is a rectangle.
46. Yes; 4 sides are ,so the opp. sides are
making it a .Since it has 4sides it is also arhombus.
47. Yes; a quad. with 4sides is a and awith 4 sides and 4right is a square.
48. 30
49. x = 5, y = 32, z = 7.5
50. x = 7.5, y = 3
51-53. Drawings mayvary. Samples aregiven.
51. Square, rectangle,isosceles trapezoid,kite.
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Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
51. (continued)
52. Rhombus, , trapezoid, kite.
53. For a < b : trapezoid, isosc.trapezoid ( a > b ), , rhombus ,kite.
For a > b : trapezoid, isos. trapezoid,, rhombus ( a < 2 b ), kite,
rectangle, square (if a = 2 b )
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Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
55. Answers may vary.Sample: only onediag. is needed.
56. Given ABCD withdiag. AC. Let AC bisect BAD .Because ABC
DAC , AB = DA byCPCTC. Butsince opp. sides of a
are , AB = CD and BC = DA.So AB = BC = CD =DA, and ABCD isa rhombus. The newstatement is true.
53. (continued)
54. a. Def. of a rhombus
b. Diagonals of abisect eachother.
c. AE AE
54. (continued)d. Reflexive Prop. of
.
e. ABC ADE
f. CPCTC
g. Add. Post.
h. AEB andAED are rt. .
i. suppl. arert. Thm.
j. Def. of
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Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
62. Answers may vary.Sample: Thediagonals of abisect each other so
AE CE . BothAED and CED are right becauseAC BD , and sinceDE DE by theReflexive Prop.,
AED CED bySAS. By CPCTC AD
CD , and sinceopp. sides of aare , AB BC AD .
57. 16, 16
58. 2, 2
59. 1, 1
60. 1, 1
61. 4. ABC ADC (ASA)
5. AB AD (CPCTC)
6. AB DC , AD BC (Opp. sides of a are .)
7. AB BC CD AD (Trans. Prop. of )
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Spec ia l Para l l e logramsSpec ia l Para l l e logramsGEOMETRY LESSON 6-4GEOMETRY LESSON 6-4
62. (continued)So ABCD is arhombus because ithas 4 sides.
63.
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Trapezo ids and Ki t esTrapezo ids and Ki t esGEOMETRY LESSON 6-5GEOMETRY LESSON 6-5
19. (continued)with KM KM bythe Reflexive Prop.means KLM
KNM by SAS. Soby CPCTC, opp.L and N are , so itis not an isos.trapezoid.
20. 12
21. 15
22. 15
23. 3
24. 4
25. 1
26. 1. ABCD is an isos. trapezoid, AB DC . (Given)
2. Draw AE DC. (Two points determine a line.)
3. AD EC (Def. of a trapezoid)
4. AECD is a . (Def. of a )
5. C 1 (Corr. are .)
6. DC AE (Opp. sides of a are .)
7. AB AE (Trans. Prop. of )
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Trapezo ids and Ki t esTrapezo ids and Ki t esGEOMETRY LESSON 6-5GEOMETRY LESSON 6-5
26. (continued)8. AEB is an isosc. . (Def. of an isosc. )
9. B 1 (Base of an isosc. are .)
10. B C (Trans. Prop. of )
11. B and BAD are suppl., C and CDAare suppl. (Same side int. are suppl.)
12. BAD CDA (Suppl. of are .)
27. 28
28. x = 35, y = 30
29. x = 18, y = 108
30. Isosc. trapezoid; allthe large rt.appear to be .
31. 112, 68, 68
32. Yes, the canbe obtuse.
33. Yes, the canbe obtuse, as wellas one other .
34. Yes; if 2 arert. , they aresuppl. The other 2are also suppl.
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Trapezo ids and Ki t sTrapezo ids and Ki t sGEOMETRY LESSON 6-5GEOMETRY LESSON 6-5
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35. No; if twoconsecutive aresuppl., then another pair must be also
because one pair of opp. is .Therefore, the opp.
would be ,which means thefigure would be aand not a kite.
36. Yes; the mustbe 45 or 135 each.
37. No; if twoconsecutive were
40. 1. AB CD , andAD CD (Given)
2. BD BD (Refl. Prop. of )
3. ABD CBD (SSS)
4. A C (CPCTC)
41. Answers may vary.Sample: Draw TAand RP
s
s
s
s
37. (continued)compl., then the kitewould be concave.
38. Rhombuses andsquares would bekites since opp.sides can be also.
39. D is any point onBN such that ND BN and D is belowN .
=/
Trapezo ids and Ki t esTrapezo ids and Ki t esGEOMETRY LESSON 6-5GEOMETRY LESSON 6-5
41. (continued)1. isosc. trapezoid TRAP (Given)
2. TA RP (Diag. of an isosc.trap. are .)
3. TR PA (Given)
4. RA RA (Refl. Prop. of )
5. TRA PAR (SSS)
6. RTA APR (CPCTC)
42. Draw BI as described, then drawBT and BP .
1. TR PA (Given)
2. R A (Base of isosc.trap. are .)
3. RB AB (Def. of bisector)
4. TRB PAB (SAS)
5. BT BP (CPCTC)
6. RBT ABP (CPCTC)
s
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Trapezo ids and Ki t esTrapezo ids and Ki t esGEOMETRY LESSON 6-5GEOMETRY LESSON 6-5
42. 7. (continued)TBI PBI
(Compl. of are .)
8. BI BI (Refl. Prop. of )
9. TBI PBI (SAS)
10. BIT BIP (CPCTC)
11. BIT and BIP are rt. .( suppl. are rt. .)
12. TI PI (CPCTC)
13. BI is bis. of TP .(Def. of bis.)
43-44. Check studentsjustifications. Samples aregiven.
43. It is one half the sum of the lengthsof the bases; draw adiag. of the trap. to form 2 . Thebases B and b of the trap. areeach a base of a . Then thesegment joining the midpts. of thenon- sides is the sum of themidsegments of the .
This sum is B + b ( B + b ).
44. It is one half the difference of the
lengths of the bases; from Ex. 43,the length of the segment joining
s
s s
s
s
12
12
12
s
Trapezo ids and Ki t esTrapezo ids and Ki t esGEOMETRY LESSON 6-5GEOMETRY LESSON 6-5
44. (continued)the midpts. of the non- sides is
(B + b ). The middle part of thissegment joins the midpts. of thediags. Each outer segmentmeasures B . So the length of the segment connecting themidpts. of the diags. is ( B b ).
45. B
46. I
47. C
48. C
49. D
50. [2]
HRW and HBW
[1] incorrect diagram OR no workshown
51. 126
52. 27
53. 27
12
12
1
2
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Trapezo ids and Ki t esTrapezo ids and Ki t esGEOMETRY LESSON 6-5GEOMETRY LESSON 6-5
54. a . 4
b. 5
c. 5
55. a. 3
b. 30
c. 30
56. SAS
Plac ing F igures in the Coord ina te P lanePlac ing F igures in the Coord ina te P laneGEOMETRY LESSON 6-6GEOMETRY LESSON 6-6
Pages 328-330 Exercises
1. W (0, h ); Z (b , 0)
2. W (a , a ); Z (a , 0)
3. W (b , b ); Z (b , b )
4. W (0, b ); Z (a , 0)
5. W ( r , 0); Z (0, t )
6. W (b , c ); Z (0, c )
7. , ;
8. a , ; undefined
9. (b , 0); undefined
10. , ;
11. ;
12. , c ; 0
13. a. (2a , 0)
b. (0, 2 b )
c. (a , b )
d. b 2 + a 2
13. (continued)e. b 2 + a 2
f. b 2 + a 2
g. MA = MB = MC
1419. Answers mayvary. Samples aregiven.
14. A, C , H , F
15. B , D , H , F
16. A, B , F , E
b 2
h 2
h b
a 2
a 2
b 2
b a
r 2
t 2
t r
b 2
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Plac ing F igures in the Coord ina te P lanePlac ing F igures in the Coord ina te P laneGEOMETRY LESSON 6-6GEOMETRY LESSON 6-6
17. A, C , G , E
18 . A, C , F , E
19. A, D , G , F
20. W (0, 2 h ); Z (2b , 0)
21. W (2a , 2 a ); Z (2a , 0)
22. W (2 b , 2 b );Z (2 b , 2 b )
23. W (0, b ); Z (2a , 0)
24. Z (0, 2 t ); W(2 r , 0)
25. W (2 b , 2c); Z (0, 2 c )
26. a. Diag. of a rhombusare .
b. Diag. of a thatis not a rhombusare not .
27. Answers may vary.Sample: r = 3, t = 2;
slopes are and ;
all lengths are 13;the opp. sides havethe same slope, sothey are . The 4
27. (continued)sides are .
28. (c a , b )
29. (a , 0)
30. (b , 0)
31. a.
b. (b , 0), (0, b ),(b , 0), (0, b )
23
23
Plac ing F igures in the Coord ina te P lanePlac ing F igures in the Coord ina te P laneGEOMETRY LESSON 6-6GEOMETRY LESSON 6-6
31. (continued)c. b 2
d. 1, 1
e. Yes, because theproduct of theslopes is 1.
32. a.
32. (continued)b.
c. b 2 + 4 c 2
d. b 2 + 4 c 2
e. the lengths are =.
33.
34. Step 1: (0, 0)
Step 2: ( a , 0)
Step 3: Since m 1 +m 2 + 90 = 180, 1and 2 mustbe compl. 3 and
2 are the acuteof a rt. .
Step 4: ( b , 0)
Step 5: ( b , a)
Step 6: Using theformula for slope, the
s
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Plac ing F igures in the Coord ina te P lanePlac ing F igures in the Coord ina te P laneGEOMETRY LESSON 6-6GEOMETRY LESSON 6-6
34. (continued)slope for 1 =
and the slope for 2= . Mult. the
slopes, = 1.
35. B
36. F
37. C
38. C
39. A
40. C
41. [2] (b , a ); the diag. of a rectangle bisect
each other.
[1] no conclusiongiven
42. 62, 118, 118; 2.5
43. (3, 2)
44. (3, 4)
45. a. Reflexive
b. AAS
b a
a b b
a a b
Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
Pages 333-337 Exercises
1. a. W , ;
Z ,
b. W (a , b );Z(c + e , d )
c. W (2a , 2 b );Z (2c + 2 e , 2 d )
d. c; it usesmultiples of 2 toname thecoordinates of W and Z .
2. a. origin
b. x -axis
c. 2
d. coordinates
3. a. y -axis
b. Distance
4. a. rt.
b. legs
4. (continued)c. multiples of 2
d. M
e. N
f. Midpoint
g. Distance
5. a. isos.
b. x -axis
c. y -axis
a 2
b 2
c + e 2
d 2
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Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
5. (continued)d. midpts.
e. sides
f. slopes
g. the DistanceFormula
6. a. (b + a )2 + c 2
b. (a + b )2 + c 2
7. a. a 2 + b 2
b. 2 a 2 + b 2
8. a. D (a b , c ),E (0, 2 c ),F (a + b , c ),G (0, 0)
b. (a + b )2 + c 2
c. (a + b )2 + c 2
d. (a + b )2 + c 2
e. (a + b )2 + c 2
f.
g.
8. (continued)h.
i.
j. sides
k. DEFG
9. a. (a , b )
b. (a , b )
c. the same point
10. Answers may vary.Sample: The
c a + b
c
a + b
c a + b
c a + b
Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
10. (continued)Midsegment Thm.;the segmentconnecting themidpts. of 2 sides of the is tothe 3rd side and half its length; you canuse the MidpointFormula and theDistance Formula toprove the statementdirectly.
11. a.
b. midpts.
11. (continued)c. (2 b , 2 c )
d. L(b , a + c ),M (b , c ), N (b , c ),K (b , a + c )
e. 0
f. vertical lines
g.
h.
1224. Answers mayvary. Samples aregiven.
12. yes; Dist. Formula
13. yes; same slope
14. yes; prod. of slopes= 1
15. no; may not haveintersection pt.
16. no; may needmeasures
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Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
17. no; may needmeasures
18. yes; prod. of slopes
of sides of A = 1
19. yes; Dist. Formula
20. yes; Dist. Formula,2 sides =
21. no; may needmeasures
22. yes; intersection pt.for all 3 segments
23. yes; slope of AB =slope of BC
24. yes; Dist. Formula,
AB = BC = CD = AD
25. 1, 4, 7
26. 0, 2, 4, 6, 8
27. 0.8, 0.4, 1.6, 2.8, 4,5.2, 6.4, 7.6, 8.8
28. 1.76, 1.52, 1.28, . . . , 9.52, 9.76
29. 2 + , 2 + 2 ,
2 + 3 , . . . . ,
2 +( n 1)
30. (0, 7.5), (3, 10),(6, 12.5)
31. 1, 6 , 1, 8 ,
(3, 10), 5, 11 ,
7, 13
32. (1.8, 6), (0.6, 7),
12n
12n
12n
12n
23
13
23
13
Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
32. (continued)(0.6, 8), (1.8, 9), (3, 10), (4.2, 11),(5.4, 12), (6.6, 13), (7.8, 14)
33. (2.76, 5.2), (2.52, 5.4),(2.28, 5.6), . . . , (8.52, 14.6),(8.76, 14.8)
34. 3, + , 5 + ,
3 + 2 , 5 + 2 , . . . . . ,
3 + ( n 1) , 5 + (n 1)
35. Assume b > a. a + ,
a + 2 , . . . . ,
a + (n 1)
36. Assume b a , d c .
a + , c + ,
a + 2 , c + 2 , . . . ,
a + (n 1) , c + (n 1)
37. a. The with bases d and b , andheights c and a , respectively,have the same area. They
12
n
10
n 12n
10n
12n
10n
b a n
b a n
b a n
b a n
d c n
b a n
d c n
b a n
d c n
s
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Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
37. a. (continued)share the small right with based and height c , and the remainingareas are with base c and
height ( b d ). So ad = bc.Mult. both sides by 2 givesad = bc .
b. The diagram shows that = ,
since both represent theslope of the top segment of the
. So by (a), ad = bc .
38. Divide the quad. into 2 . Find the
centroid for each and connectthem. Now divide the quad. into 2
38. (continued)other and follow the samesteps. Where the two lines meetconnecting the centroids of the 4
is the centroid of the quad.
39. a. L(b , d ), M (b + c , d ), N (c , 0)
b. AM : y = x ;
BN : y = (x c );
CL: y = (x 2c )
c. P ,
d. Pt. P satisfies the eqs. for AM
s
12 12
a b
c d
s
s
s
d b + c
2d 2b c
2(b + c )3
2d 3
d b 2c
Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
39. d. (continued)and CL.
e. AM = (b + c )2 + d 2 ;
AP = =
(b + c )2 + d 2 =
(b + c )2 + d 2 = AM
The other 2 distances are foundsimilarly
40. a.
40. (continued)b. Let a pt. on line p be ( x , y ).
Then the eq. of p is =
or y = (x a ).
c. x = 0
d. When x = 0, y = (x a ) =
(a ) = . So p and q
intersect at 0, .
e.
f. Let a pt. on line r be ( x , y ).
Then the eq. of r is =
2(b + c ) 23
2d 23
2 2
3
23
23
b c
y 0x a
b c b
c
ab c
b c
a c
ab c
y 0x b
a c
b c
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Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
40. f. (continued)or y = (x b ).
g. = (0 b )
h. 0,
41. a. Horiz. lines have slope 0, andvert. lines have undef. slope.Neither could be mult. to get 1.
b. Assume the lines do notintersect. Then they have thesame slope, say m . Then m m =m 2 = 1, which is impossible. Sothe lines must intersect.
41. (continued)c. Let the eq. for 1 be y = x
and for 2 be y = x and the
origin be the int. point.
a c
ab c
a c
ab c
b a
a b
Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
41. c. (continued)Define C (a , b ), A(0, 0), and
B a , . Using the Distance
Formula, AC = a 2 + b 2,
BA = a 2 + , and
CB = b + .
Then AC 2 + BA2 = CB 2, andm A = 90 by the Conv. of thePythagorean Thm. So 1 2 .
42. A
43. G
44. [2] a. = 3; a = 1;
= 4; b = 11;
(a , b ) = (1, 11)
b. (7 (1)) 2 + (3 11) 2
= 260 = 2 65
16.12
[1] minor computational error ORno work shown
a 2 b
a 4b 2
a 2b
7 + a 2
3 + b 2
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Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
45. [4] a-b. Sample:
c. AP = (b a )2 + c 2 = RQ PQ = PQ AQ = a = RP
APQ RQP by SSS[3] minor computational error
[2] parts (a) and (b) correct
[1] one part correct
46. (a , b )
47. a. If the sum of the of a polygonis 360, then the polygon is a
quad.
b. If a polygon is a quad., then thesum of its is 360.
48. a. If x 51, then 2 x 102.
b. If 2x 102, then x 51.
49. a. If a 5, then a 2 25.
b. If a 2 25, then a 5.
s
s
=/ =/
=/ =/
=/ =/
=/ =/
Proofs Us ing Coord ina te Geomet ryProofs Us ing Coord ina te Geomet ryGEOMETRY LESSON 6-7GEOMETRY LESSON 6-7
50. a. If b 4, then b is not neg.
b. If b is not neg., then b 4.
51. a . If c 0, then c is not pos.
b. If c is not pos., then c 0.
52. A C , AD CD and ADB CDB so by ASA ADB
CDB and by CPCTC AB CB .
53. HE FG , EF GH , and HF HF by the Refl. Prop. of , so
HEF FGH by SSS. ThenCPCTC 1 2.
54. LM NK , LN NL by the Refl.Prop. of , and LNK NLM by all rt. are . So LNK
NLM by SAS, and K M byCPCTC.
s