Download CHAPTER 5

CHAPTER 5
Chapter Opener
20. B; this statement matches the corresponding vertices
in order.
Chapter Readiness Quiz (p. 232)
1. D
2. G
3. D
21. a R a N
22. LN
**** PR
****
23. a RPQ a NLM
24. QP
**** ML
****
25. T MNL T QRP
Lesson 5.1
5.1 Checkpoint (pp. 234–235)
1. Corresponding Angles
Corresponding Sides
aS aY
ST
**** YX
****
aU aZ
SU
**** YZ
****
aT aX
UT
ZX
**** ****
28. no
29. yes
30. yes
31. no
32. no
33. Corresponding Angles
2. Because DF
**** ****
AC , you know that DF AC 3.
4. Yes; the congruent sides are marked on the diagram, so
XY
ZW , XV
**** **&
**** ZV
**** , and YV
**** WV
**&.
Vertical angles are congruent.
aX aZ
Alternate Interior Angles Theorem
aY aW
Alternate Interior Angles Theorem
aP aS
QP
**** TS
****
aQ aT
QR
**** TU
****
aR aU
PR SU
****
****
34. Corresponding Angles
in order.
a XVY a ZVW
AC EF
****
****
aC aF
AB ED
****
****
aB aD
CB
**** FD
****
Since all corresponding parts are congruent,
T ABC T EDF.
35.
5.1 Guided Practice (p. 236)
A
1. corresponding angles
2. neither
4. neither
5. a F
6. a Z
7. ED
****
8. YZ
****
9. ma P ma L 105
F
B
E
A
C
ma B ma E 100
12. LN PR 8.5
13. Yes; the congruent sides are marked on the diagram,
so FE
**** LK
****, FG
**** ****
LJ , and GE
**** JK
****. Two pairs of
congruent angles are also marked on the diagram, so
a E a K and a G a J. The third pair of corresponding angles are right angles and are congruent since all
right angles are congruent. All corresponding parts are
congruent, so T EFG T KLJ.
5.1 Practice and Applications (pp. 236–239)
14. corresponding angles
15. neither
16. neither
17. corresponding sides
18. corresponding sides
19. corresponding angles
D
B
37. DF AC 5;
11. QR MN 11.6
Copyright © McDougal Littell Inc.
All rights reserved.
36.
C
F
10. ma M ma Q 45
Corresponding Sides
aA aE
Since all corresponding parts are congruent,
T XYV T ZWV.
3. corresponding sides
Corresponding Sides
Since all corresponding parts are congruent,
T QPR T TSU.
Because a B a E, you know that ma B ma E 28.
3. B; This statement matches up the corresponding vertices
26. T PRQ T LNM
27. yes
39. AB DE 6;
ma F ma C 50
E
D
38. AB DE 14;
ma F ma C 39
40. BC EF 5;
ma D ma A 27
41. C; this statement does not match up the corresponding
vertices in order.
42. Corresponding Angles
Corresponding Sides
aA aD
AB DF
****
****
aB aF
AC DE
****
****
aC aE
CB
**** EF
****
Since all corresponding parts are congruent,
T ABC T DFE.
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
69
Chapter 5 continued
43. Corresponding Angles
Corresponding Sides
aJ aP
JK
**** PN
****
aK aN
JL PM
****
**&
aL aM
KL
NM
**** **&
Since all corresponding parts are congruent,
T JKL T PNM.
58. obtuse; 62 52 22
59. acute; 102 52 92
60. obtuse; 142 52 122
5.1 Algebra Skills (p. 239)
61. 11.95
62. 3.72
63. 19.33
64. 0.74
65. 17.076
66. 3.222
44. The triangles are not congruent.
Lesson 5.2
45. Corresponding Angles
a BCA a DCE
Vertical Angles Theorem
aB aD
Alternate Interior Angles Theorem
aA aE
Alternate Interior Angles Theorem
5.2 Activity (p. 240)
1. All triangles made with two pencils and a 45 angle
appear to be congruent.
2. You need to show that the angle formed by the two
Corresponding Sides
known sides of one triangle is congruent to the angle
formed by the two known sides of the other triangle.
BC
**** DC
****
AC EC
****
****
5.2 Checkpoint (p. 244)
AB ED
****
****
Since all corresponding parts are congruent,
1. Statements
Reasons
T ABC T EDC.
1. CB
**** CE
****
1. Given
46. Corresponding Angles
2. ****
AC DC
****
2. Given
Corresponding Sides
a LJM a JLK
JM
**** LK
****
3. a BCA a ECD
3. Vertical Angles Theorem
a JLM a LJK
ML
**** KJ
****
4. T BCA T ECD
4. SAS Congruence Postulate
aM aK
JL ****
****
LJ
5.2 Guided Practice (p. 245)
Since all corresponding parts are congruent,
T JML T LKJ.
1. a JKM
2. a J
47. The triangles are not congruent.
4. a KML
5. a LKM
48. Perpendicular lines intersect to form four right angles,
6. Yes; T ABC T DEC by the SAS Congruence Postulate
and all right angles are congruent.
49. By the Base Angles Theorem, the base angles of an
isosceles triangle are congruent, so a JGF a JHF.
50. Yes; by the Reflexive Property of Congruence, FJ
**** FJ
****
so all three pairs of corresponding sides are congruent.
By Exercises 48 and 49, all three pairs of corresponding
angles are congruent.
51. a BAC and a DAE are congruent because of the
Reflexive Property of Congruence.
52. No; the corresponding angles are congruent, but the
corresponding sides are not congruent. That is,
ED
AB .
**** ç CB
**** , AE
**** ç AC
**** , and AD
**** ç ****
5.1 Standardized Test Practice (p. 239)
53. B
5.1 Mixed Review (p. 239)
54. Reflexive Property of Equality
55. Reflexive Property of Congruence
56. Symmetric Property of Congruence
57. Transitive Property of Equality
70
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
3. a L
because ****
AC DC
**** , a ACB a DCE, and BC
**** EC
**** .
7. No; there is nothing in the diagram indicating that
a FHG a JHK or that FG
**** JK
**** .
8. Yes; T PQR T SRQ by the SSS Congruence Postulate.
By the Reflexive Property of Congruence, RQ
**** RQ
**** , so
all three pairs of corresponding sides are congruent.
5.2 Practice and Applications (pp. 245–249)
9. a ABD
12. a A
10. a C
13. a BDC
11. a C
14. a DBC
15. Yes; T ABC T DBE because all three pairs of corre-
sponding sides of the triangles are congruent.
16. Yes; by the Reflexive Property of Congruence, ****
JL ****
JL
so T JKL T JML because all three pairs of corresponding sides of the triangles are congruent.
17. No; there is no information about PS
**** and RS
****.
18. Yes; by the Reflexive Property of Congruence, SQ
**** SQ
****
so T PQS T RQS because two sides and the included
angle of T PQS are congruent to two sides and the
included angle of T RQS.
19. No; the congruent angles are not included by the
congruent sides.
Copyright © McDougal Littell Inc.
All rights reserved.
Chapter 5 continued
20. Yes; by the Vertical Angles Theorem, a JNK a MNL
so T JNK T MNL because two sides and the included
angle of T JNK are congruent to two sides and the
included angle of T MNL.
21. No; the congruent angles are not included by the
congruent sides.
22. Yes; T GHK T JKH by the SAS Congruence
Postulate because GH
**** JK
**** (Given), a GHK a JKH
(Alternate Interior Angles Theorem), and KH
**** KH
****
(Reflexive Property of Congruence).
23. Yes; T MNP T PQM by the SSS Congruence Postulate
because MN
**& PQ
**** (Given), **&
MQ PN
**** (Given), and
**& (Reflexive Property of Congruence).
MP
**& MP
24. Yes; T WXZ T YXZ by the SAS Congruence Postulate
because WX
XZ ****
XZ (Reflexive
**** YX
**** (Given), and ****
Property of Congruence).
36. Statements
1. ****
AC BC
****
AB and CB
26. No; there is no information about ****
****.
AB CD
27. ****
****
28. a ACB a CED
29. To use the SAS Congruence Postulate, you need to know
that BC
**** EF
****.
30. To use the SSS Congruence Postulate, you need to know
that JK
**** ML
**** . To use the SAS Congruence Postulate,
you need to know that a JLK a MKL.
31. To use the SSS Congruence Postulate, you need to know
that PQ
**** RQ
**** . To use the SAS Congruence Postulate,
you need to know that a PSQ a RSQ.
32. Sample answer: Since WXYZ is a square, all four sides
of WXYZ are congruent. ****
XZ ****
XZ by the Reflexive
Property of Congruence, so TWXZ TYXZ by the SSS
Congruence Postulate.
AB CD
33. Sample answer: Since ****
**** , a ABC a DCB
AB DC
(Alternate Interior Angles Theorem). ****
**** (Given)
and CB
**** CB
**** (Reflexive Property of Congruence), so
T ABC T DCB by the SAS Congruence Postulate.
1. Given
2. M is the midpoint of ****
AB. 2. Given
3. AM
**& BM
****
3. Definition of midpoint
4. CM
**& CM
**&
4. Reflexive Prop. of Cong.
5. T ACM T BCM
5. SSS Cong. Postulate
37. Maria should not have used the SAS Congruence
Postulate because the congruent angles are not included
by the congruent sides.
38. Statements
Reasons
1. T ABD and T CBD
are equilateral.
1. Given
2. BD
**** BD
****
2. Reflexive Prop. of Cong.
3. BC
**** BD
**** ,
3. Definition of equilateral
25. Yes; T RSU T TSU by the SSS Congruence Postulate
because RS
**** TS
**** (Given), RU
**** TU
**** (Given), and
SU
**** SU
**** (Reflexive Property of Congruence). Also,
T RSU T TSU by the SAS Congruence Postulate
because RS
**** TS
**** (Given), a SRU a STU (Base Angles
Theorem), and RU
**** TU
**** (Given).
Reasons
BA BD
****
****
4. BC
BA
**** ****
4. Transitive Prop. of Cong.
5. DC
**** BD
**** ,
5. Definition of equilateral
DA
**** BD
****
6. DC
**** DA
****
7. T ABD T CBD
39. Statements
6. Transitive Prop. of Cong.
7. SSS Cong. Postulate
Reasons
1. XZ
**** YZ
****
1. Given
2. **&
ZM bisects a YZX.
2. Given
3. a YZM a XZM
3. Def. of angle bisector
4. **&
ZM **&
ZM
4. Reflexive Prop. of Cong.
5. T ZMY T ZMX
5. SAS Cong. Postulate
5.2 Standardized Test Practice (p. 248)
40. C
41. G
5.2 Mixed Review (p. 248)
42. alternate exterior angles
43. same-side interior angles
44. corresponding angles
45. alternate interior angles
46. corresponding angles
47. same-side interior angles
Reasons
48. no; 14 8 25
1. EF
**** GH
****
1. Given
49. yes; 20 10 28, 20 28 10, and 10 28 20
2. FG
**** HE
****
2. Given
50. no; 16 14 30
3. GE
**** GE
****
3. Reflexive Prop. of Cong.
5.2 Algebra Skills (p. 249)
4. T EFG T GHE
4. SSS Congruence Postulate
34. Statements
35. Statements
Reasons
1. SP
**** TP
****
1. Given
2. PQ
**** bisects a SPT.
2. Given
3. a SPQ a TPQ
3. Def. of angle bisector
4. PQ
**** PQ
****
4. Reflexive Prop. of Cong.
5. T SPQ T TPQ
5. SAS Cong. Postulate
Copyright © McDougal Littell Inc.
All rights reserved.
1.7
51. 3
52. 1
2
3.5
53. 4
0
6.3
54. 1
5
9
12.6
55. 1
4
.7
6
3.8
56. 0
.8
7
0.9
57. 1
.1
2
1.1
58. 4
0
.8
5
6.4
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
71
Chapter 5 continued
11. ****
AC is included between a BAC and a BCA.
Quiz 1 (p. 249)
1. a R a C
2. ****
AB QP
****
12. ****
AC is not included between a DAC and a D.
3. T BAC T PQR
4. T RPQ T CBA
13. BC
**** is not included between a CAB and a B.
5. T FEG T XYZ, T EGF T YZX, or T GFE T ZXY
6. No; the congruent angles are not included by the
congruent sides.
7. Yes; by the Reflexive Property of Congruence, FG
**** FG
**** .
The three sides of T DGF are congruent to the three sides
of T EGF, so T DGF T EGF by the SSS Congruence
Postulate.
8. Yes, two sides and the included angles of T JKF are
congruent to two sides and the included angle of T NMF,
so the triangles are congruent by the SAS Congruence
Postulate.
14. ASA Congruence Postulate; two angles and the included
side of T ABC are congruent to two angles and the
included side of T DEF.
15. AAS Congruence Theorem; two angles and a non-
included side of T JLM are congruent to two angles
and the corresponding non-included side of T PNM.
16. ASA Congruence Postulate; two angles and the included
side of T URS are congruent to two angles and the
included side of T UTS.
17. Yes; SAS Congruence Postulate; two sides and the
included angle of T PQR are congruent to two sides
and the included angle of T TSM.
18. Yes; AAS Congruence Theorem; a A a D (Given),
Lesson 5.3
5.3 Geo-Activity (p. 250)
Step 5. The triangles are congruent.
a BCA a BCD (Perpendicular lines form right angles,
and all right angles are congruent.), and BC
**** BC
****
(Reflexive Property of Congruence). So,
T BCA T BCD.
19. Yes; ASA Congruence Postulate; a S a T (Given),
5.3 Checkpoint (p. 253)
1. You can use the ASA Congruence Postulate when
the corresponding sides are included between the
corresponding angles.
2. No; based on the diagram, you know only that
a BAC a DCA (Alternate Interior Angles Theorem)
and ****
AC ****
AC .
3. No; there is no information about any of the sides.
4. Yes; T KLP T MLN by the AAS Congruence Theorem
because a P a N (Given), a KLP a MLN (Vertical
Angles Theorem), and KL
**** ML
**** (Given).
5.3 Guided Practice (p. 253)
1. FG
**** is included between a F and a G.
2. GH
**** is not included between a F and a G.
SV
**** TV
**** (Given), and a SVR a TVU (Vertical Angles
Theorem). So, T SVR T TVU.
20. No; from the diagram, you know only that EF
JH ,
**** ****
FG
**** HG
**** , and a EGF a JGH (Vertical Angles
Theorem). The congruent angles are not included by the
congruent sides.
21. Yes; SSS Congruence Postulate; WX
ZX (Given),
**& ****
WY
XY ****
XY (Reflexive Property of
**& ZY
**** (Given), and ****
Congruence). So, T XWY T XZY.
22. No; from the diagram, you know only that NL
**** NL
****
(Reflexive Property of Congruence) and a MNL a KLN
(Alternate Interior Angles Theorem).
23. AAS Congruence Theorem; two angles and a non-
included side of T JKL are congruent to two angles
and the corresponding non-included side of T PQR.
3. FH
**** is not included between a H and a G.
24. AC
**** FD
****
25. a K a Q
4. HG
**** is included between a H and a G.
27. a C a D
28. a CAB a DAB
5. ****
AB DE
****
30. BC
**** BD
****
6. a A a D
7. Yes; T RST T TQR by the ASA Congruence Postulate
26. a WZX a YXZ
29. AC
**** AD
****
31. Sample answer:
because a STR a QRT (Given), RT
**** RT
**** (Reflexive
Property of Congruence), and a SRT a QTR (Given).
T
8. Yes; T JKL T NML by the AAS Congruence Theorem
because a KLJ a MLN (Vertical Angles Theorem),
a JKL a NML (Given), and ****
JK NM
**& (Given).
9. No; you would either need to know that a A a D or
P
R
S
U
32. Sample answer:
a C a F.
5.3 Practice and Applications (pp. 254–256)
T
P
R
S
U
AB is not included between a B and a BCA.
10. ****
72
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
Copyright © McDougal Littell Inc.
All rights reserved.
Chapter 5 continued
7
9
P
2
7
7
9
7
7
9
2
7
2
49
18
13
18
50. 2
33. Sample answer:
S
12
25
3
10
3
10
25
12
1
3 25 5
2 10 12 4
5
8
51. R
T
U
34. Statements
3
4
4
3
8
4
1
3
1. GF
**** GL
****
1. Given
2. FH
**** LK
****
2. Given
3. a F a L
3. Alt. Int. Angles Theorem
4. a HGF a KGL
4. Vertical Angles Theorem
Lesson 5.4
5. T FGH T LGK
5. ASA Cong. Postulate
5.4 Checkpoint (p. 259)
35. Statements
Reasons
1. BC
**** EC
****
32
3
2
3
52. 8 8 10
Reasons
4
11
4
11
1
12
1
4
1
11 12 3
1
33
53. 12 1. Yes; HL Congruence Theorem; the hypotenuse and a leg
2. ****
AB ∏ AD
****
2. Given
of right T BAC are congruent to the corresponding
hypotenuse and a leg of right T DEC.
3. DE ∏ AD
3. Given
2. No; from the diagram, you know only that SU
**** SU
****
4. a A and a D
are right angles.
4. Perpendicular lines form
right angles.
5. a A a D
5. Right angles are congruent.
6. a ACB a DCE
6. Vertical Angles Theorem
7. T ABC T DEC
7. AAS Congruence Theorem
1. Given
5.3 Standardized Test Practice (p. 256)
(Reference Property of Congruence) and a RSU a TUS
(Alternate Interior Angles Theorem).
3. Yes; SSS Congruence Postulate, SAS Congruence
Postulate, or HL Congruence Theorem
5.4 Guided Practice (p. 260)
1. hypotenuse
2. leg
3. leg
4. leg
36. D
5. hypotenuse
6. leg
5.3 Mixed Review (p. 256)
7. Yes; all three sides of T EDG are congruent to the three
37. 102 b2 182
38. c2 112 62
100 b2 324
c2 121 36
b2 224
c2 157
b 224
c 157
b 15.0
c 12.5
39. 15 a 26
2
2
2
a2 451
gruent to the hypotenuse and a leg of right T FED.
40. a C a F
41. BA
**** ED
****
42. a D a A
43. EF
**** BC
****
44. a B a E
45. CB
**** FE
****
5.3 Algebra Skills (p. 256)
5
16
1
3
26 2
3
2
1
1
4
48. 5
8
16
5
11. Yes; ****
JL ****
JL (Reflexive Property of Congruence), so the
hypotenuse and a leg of right T JKL are congruent to the
hypotenuse and a leg of right T JML.
12. No; there is no information given about the lengths of the
1
1
1
1
1
1
46. 6 12
2
2
6
2
6
4
4
1
4
1
1
47. 4 5
5
4
5
4
5
5
8
9. No; both triangles are equilateral and equiangular, but
10. Yes; the hypotenuse and a leg of right T ABC are con-
a 21.2
1
6
congruent to the hypotenuse and a leg of right T QPM,
so the triangles are congruent by the HL Congruence
Theorem.
5.4 Practice and Applications (pp. 260–263)
a 451
2
3
8. Yes; the hypotenuse and a leg of right T MNQ are
there is no information about the lengths of the sides of
the triangles.
225 a2 676
1
6
sides of T GFE so the triangles are congruent by the SSS
Congruence Postulate.
hypotenuses of the right triangles.
13. ****
JL **&
ML
14. SAS Congruence Postulate; since K is the midpoint of
JM
**** , JK
**** **&.
MK It is given that a JKL and a MKL are
right angles, and all right angles are congruent. By the
Reflexive Property of Congruence LK
**** LK
****.
1
5 16 2
1 8 51
49. 2
Copyright © McDougal Littell Inc.
All rights reserved.
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
73
Chapter 5 continued
15. Yes; SAS Congruence Postulate; two sides and the
27. Sample answer:
included angle of T ABC are congruent to two sides
and the included angle of T DEF.
N
U
16. Yes; ASA Congruence Postulate; two angles and the
included side of T JKL are congruent to two angles
and the included side of T ZYX.
17. No; the congruent angles are not included between the
congruent sides, so there is not enough information to
prove that the triangles are congruent.
18. Yes; SSS Congruence Postulate; FH
**** FH
**** (Reflexive
Property of Congruence), so all three sides of T FGH
are congruent to the three sides of T FJH.
19. Yes; AAS Congruence Theorem; perpendicular lines
form right angles, and all right angles are congruent, so
a UTS a UTV. Also, by the Reflexive Property of
Congruence, UT
**** UT
**** . Two angles and a non-included
side of T STU are congruent to two angles and the corresponding non-included side of T VTU.
20. Yes; AAS Congruence Theorem; perpendicular lines
form right angles, and all right angles are congruent, so
a A a D. Also, vertical angles are congruent, so
a ACB a DCE. Two angles and a non-included side of
T ABC are congruent to two angles and the corresponding
non-included side of T DEC.
21. Yes; HL Congruence Theorem; by the Reflexive Property
of Congruence, DB
**** DB
**** , so the hypotenuse and a leg
of right T ADB are congruent to the hypotenuse and corresponding leg of right T CDB.
22. Yes; ASA Congruence Postulate; a JKL a MLK
and a JLK a MKL by the Alternate Interior Angles
Theorem. LK
**** LK
**** by the Reflexive Property of
Congruence. So, two angles and the included side of
T JKL are congruent to two angles and the included
side of T MLK.
23. Yes; SAS Congruence Postulate; DF
**** DF
**** (Reflexive
Property of Congruence), so two sides and the included
angle of T CDF are congruent to two sides and the
included angle of T EDF.
24. Meghan is correct because it is given that AB
**** CD
**** and
BC
**** DA
**** , and by the Reflexive Property of Congruence,
AC ****
AC . Keith is correct because two sides and the
****
included angle of T ABC are congruent to two sides and
the included angle of T CDA. Angle is correct because
the hypotenuse (AC
**** ) and a leg of right T ABC are congruent to the hypotenuse and a leg of right T CDA.
25. Sample answer:
L
T
N
U
M
S
26. Sample answer:
L
74
L
M
T
S
28. Sample answer:
M
T
L
N
S
U
29. a F a J, ASA Congruence Postulate; or a G a L,
AAS Congruence Theorem; or GH
**& LK
**** , SAS
Congruence Postulate.
30. PQ
**& RS
**** or PS
**** RQ
**** , HL Congruence Theorem; or
a PQS a RSQ or a PSQ a RQS, AAS Congruence
Theorem
31. WV
**& ZX
**** , ASA Congruence Postulate; or VW
**& YZ
**** , or
VX
**** YX
**** , AAS Congruence Theorem
32. Statements
Reasons
1. BD
**** FD
****
1. Given
2. D is the midpoint
of CE
**** .
2. Given
3. CD
**** ED
****
3. Definition of midpoint
4. a BCD and a FED
are right angles.
4. Given
5. T BCD and T FED
are right triangles.
5. Definition of right triangle
6. T BCD T FED
6. HL Congruence Theorem
5.4 Standardized Test Practice (p. 262)
33. a. Sample answer: BD
**** BD
**** by the Reflexive Property
of Congruence; the hypotenuse and a leg of right
T ABD are congruent to the hypotenuse and a leg of
right T CBD, so T ABD T CBD by the HL
Congruence Theorem.
b. Yes; Sample answer: It is given that AB
**** BC
**** .
By the Base Angles Theorem, a BAD a BCD.
Perpendicular lines form right angles, and all right
angles are congruent, so a BDA a BDC. Therefore,
T ABD T CBD by the AAS Congruence Theorem.
c. Yes; Sample answer: You know that T CEG T FEG by the HL Congruence Theorem. It is given
that the hypotenuse and a leg of right T ABD are to
the hypotenuse and a leg of right T CEG. So,
TABD TCEG by the HL Congruence Theorem.
Since TABD TCBD, all four right triangles are
congruent.
5.4 Mixed Review (p. 263)
N
U
M
T
34. ma 1 110 by the Alternate Interior Angles Theorem;
S
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
ma 2 180 110 70 by the Same-Side Interior
Angles Theorem (or by the Linear Pair Postulate).
Copyright © McDougal Littell Inc.
All rights reserved.
Chapter 5 continued
35. ma 1 82 by the Corresponding Angles Postulate;
10. Yes; SSS Congruence Postulate; T JML and T JKL are
ma 2 82 by the Alternate Exterior Angles Theorem
(or by the Vertical Angles Theorem).
equiangular and therefore equilateral. Because they share a
common side (JL
**** ) you know that all the sides of T JML
are congruent to all the sides of T JKL; or AAS
Congruence Theorem because ****
JL ****
JL (Reflexive
Property of Congruence), two angles and a non-included
side of T JML are congruent to two angles and the corresponding non-included side of T JKL; or ASA Congruence
Postulate because ****
JL ****
JL , two angles and the included
side of T JML are congruent to two angles and the
included side of T JKL.
36. ma 1 180 57 123 by the Linear Pair Postulate;
ma 2 57 by the Alternate Interior Angles Theorem.
37. Yes; AAS Congruence Theorem; two angles and a non-
included side of T ABC are congruent to two angles and
a non-included side of T DEF.
38. No; a PQS a RSQ by the Alternate Interior Angles
Theorem and SQ
**** SQ
**** by the Reflexive Property of
Congruence, but the congruent angles are not included
between the congruent sides.
39. Yes; AAS Congruence Theorem; a JLK a NLM by
the Vertical Angles Theorem, so two angles and a nonincluded side of T JKL are congruent to two angles and
the corresponding non-included side of T NML.
5.4 Algebra Skills (p. 263)
40. 2 4 5 8 5 13
5.4 Technology Activity (p. 264)
1. Answers will vary, but BG BH.
2. ****
AB ****
AB and BG
**** BH
****
3. a BAG a BAH
4. No; Sample answer: T ABH and T ABG have different
shapes.
5.
41. 10 5 2 10 10 0
42. 3 42 11 3 16 11 19 11 8
43. 7 2 6 3 14 18 32
A
NM
**& PM
**& (Given), and a LNM a QPM (All right
angles are congruent.).
2. Yes; a NML a PMQ (Vertical Angles Theorem),
H
Lesson 5.5
5.5 Checkpoint (p. 267)
1. SAS Congruence Postulate; AD
**** CB
**** (Given),
a DAC a BCA (Alternate Interior Angles Theorem),
and ****
AC ****
AC (Reflexive Property of Congruence). So,
T ADC T CBA and the corresponding parts ****
AB and
CD
**** are congruent.
a LNM a QPM (All right angles are congruent.) and
LN
**** QP
**** (Given).
3. No; no information is given about the lengths of the
hypotenuses.
A
sible for two sides and a non-included angle of one triangle to be congruent to two sides and the corresponding
non-included angle of another triangle without the two
triangles being congruent.
45. 52 10 2 25 20 5
1. Yes; a NML a PMQ (Vertical Angles Theorem),
G
6. There is no SSA Congruence Postulate because it is pos-
44. 3 5 2 7 15 14 1
Quiz 2 (p. 263)
B
B
2.
K
K
4. No; no information is given about the third sides.
N
5. No; the congruent sides are not corresponding sides.
6. Yes; AAS Congruence Theorem; two angles and a non-
J
included side of T JKL are congruent to two angles and
the corresponding non-included side of T QRP.
8. Yes; HL Congruence Theorem; the hypotenuse and a leg
of right T HEF are congruent to the hypotenuse and a
right leg of T FGH.
9. Yes; AAS Congruence Theorem; a RTS a UTV
L
T JKN T LKM by the ASA Congruence Postulate,
so the corresponding parts NJ
ML are congruent.
**** and **&
7. Yes; ASA Congruence Postulate; TU
**** TU
**** by the
Reflexive Property of Congruence, so two angles and the
included side of T STU are congruent to two angles and
the included side of T VUT.
M
3.
S
P
R
P
R
In the original diagram, PR
**** and PR
**** are the same side,
so PR
**** PR
**** . T PQR T RSP by the AAS Congruence
Theorem.
(Vertical Angles Theorem), a RST a VUT (Alternate
Interior Angles Theorem), and RS
**** UV
**& (Given).
So, two angles and a non-included side of T RST are
congruent to two angles and the corresponding
non-included side of T UVT.
Copyright © McDougal Littell Inc.
All rights reserved.
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
75
Chapter 5 continued
5.5 Guided Practice (p. 268)
15. Statements
Reasons
1. Corresponding parts of congruent triangles are congruent.
1. AD
**** CD
****
1. Given
2. B
2. a ABD and a CBD
are right angles.
2. Given
3. T ABD T CBD
are right triangles.
3. Def. of right triangle
4. BD
**** BD
****
4. Reflexive Property of Cong.
5. T ABD T CBD
5. HL Congruence Theorem
6. a A a C
6. Corresponding parts of triangles are .
3. When two parallel lines are cut by a transversal, alternate
interior angles are congruent, so a VSU a TUS and
a TSU a VUS. Also, SU
**** SU
**** by the Reflexive
Property of Congruence. T STU T UVS by the ASA
Congruence Postulate and a STU a UVS since corresponding parts of congruent triangles are congruent.
4. a A a D, a DBC a ACB, and BC
**** BC
**** . So,
T ABC T DCB by the AAS Congruence Theorem
and ****
AB DC
**** since corresponding parts of congruent
triangles are congruent.
5. LN
**** LK
**** , a LNM a LKJ, and a L a L. So,
T JKL T MNL by the ASA Congruence Postulate
and a J a M since corresponding parts of congruent
triangles are congruent.
5.5 Practice and Applications (pp. 268–271)
6. Corresponding parts of congruent triangles are congruent.
7. T ABC T DBC
8. T JKL T NML
9. T EDF T ABC
10. B
D
A
C
In the original diagram, DB
**** and DB
**** are the same side,
so DB
**** DB
**** . T ABD T CDB by the SAS Congruence
Postulate.
11. E
J
G
G
12. AAS Congruence Theorem; a C a F, a CBA a FED,
and CA
**** FD
**** , so T ABC T DEF.
13. ASA Congruence Postulate; a GJH a KMH and
HJ
**** HM
**&. By the Reflexive Property of Congruence,
a H a H. So, T GHJ T KHM.
14. Statements
1. Given
2. a JKL a NML
2. Alternate Interior Angles
Theorem
3. KL
**** ML
****
3. Given
4. a JLK a NLM
4. Vertical Angles Theorem
5. T JKL T NML
5. ASA Congruence Postulate
6. ****
JK NM
**&
6. Corresponding parts of triangles are .
Reasons
Reasons
1. a TRS a RTQ
1. Given
2. a RST and a TQR
are right angles.
2. Given
3. a RST a TQR
3. All right angles are .
4. RQ
**** TS
****
4. Given
5. T RQT T TSR
5. AAS Congruence Theorem
6. QT
**** SR
****
6. Corresponding parts of triangles are .
H
In the original diagram, a G and a G are the same angle,
so a G a G. T EFG T HJG by the AAS
Congruence Theorem.
Reasons
1. JK
NM
**** **&
17. Statements
D B
F
16. Statements
18. Statements
Reasons
1. UV
XY
**** ****
1. Given
2. a UVW a XYZ
2. Given
3. VW
**& YZ
****
3. Given
4. T UVW T XYZ
4. SAS Congruence Postulate
5. WU
**& ZX
****
5. Corresponding parts of triangles are .
19. Statements
Reasons
1. ****
AB AE
****
1. Given
2. a ACB a ADE
1. BD
**** and AE
**** bisect
each other at C.
1. Given
2. Given
3. a A a A
3. Reflexive Prop. of Cong.
2. BC
**** DC
****
2. Def. of segment bisector
4. T ABC T AED
4. AAS Congruence Theorem
3. ****
AC EC
****
3. Def. of segment bisector
5. a B a E
5. Corresponding parts of triangles are .
4. a BCA a DCE
4. Vertical Angles Theorem
5. T ABC T EDC
5. SAS Congruence Postulate
6. a A a E
6. Corresponding parts of triangles are .
76
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
Copyright © McDougal Littell Inc.
All rights reserved.
Chapter 5 continued
20. Statements
Reasons
1. ****
JL **&
MK
1. Given
2. JK
**** ∏ LK
**** , ML
**** ∏ KL
****
2. Given
3. a JKL and a MLK
are right angles.
3. Perpendicular lines form
right angles.
4. T JKL and T MLK
are right triangles.
4. Definition of right triangle
5. KL
**** KL
****
5. Reflexive Prop. of Cong.
6. T JKL T MLK
6. HL Congruence Theorem
7. JK
**** ML
****
7. Corresponding parts of triangles are .
21. ASA Congruence Postulate; a LJK a PMN (Given),
JL **&
MP (Segment Addition Postulate), and a JLK ****
a MPN (Corresponding Angles Theorem). Two angles
and the included side of T JKL are congruent to two
angles and the included side of T MNP.
29.
8x 2 2 90 2
8x 88
8x
88
8
8
x 11
5.5 Algebra Skills (p. 271)
30.
x3
31. 7x 63
7x
63
7
7
x 9
32.
4x 32
4x
32
4
4
x8
23. G
(x 3) 25
x 3 3 25 3
4x 9 23
4x 9 9 23 9
5.5 Mixed Review (p. 271)
24.
x58
x5585
5.5 Standardized Test Practice (p. 271)
22. A
(8x 2) 90
3x 8 19
2x 10 10 24 10
3x 8 8 19 8
2x 14
2x
14
2
2
x7
(6x 1) (5x 5)
x1151
x6
27.
x 48 90
28. (4x 1) 37 90
5.6 Activity (p. 272)
1. CM
^&( is perpendicular to ****
AB and CM
^&( bisects ****
AB .
2. Answers will vary, but CA CB, DA DB, EA EB,
and FA FB.
3. Any point on the perpendicular bisector of a segment is
the same distance from the endpoints of the segment.
5.6 Checkpoint (p. 274)
1.
FH HG
4x 38 90
x 3 2x 1
4x 38 38 90 38
x 3 x 2x 1 x
4x 52
4x
52
4
4
x 13
Copyright © McDougal Littell Inc.
All rights reserved.
3x 27
3x
27
3
3
x9
Lesson 5.6
x 48 48 90 48
x 42
3x 21
3x
21
3
3
x7
2x 10 24
3x
54
3
3
x 18
x15
11 3x 11 32 11
35. x 2x 8 19
25. 3x 54
6x 1 5x 5x 5 5x
11 3x 32
34. 5x 3x 10 24
x 28
26.
33.
3x1
31x11
2x
FH x 3 2 3 5
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
77
Chapter 5 continued
9. Paige cannot assume that PC
&
*(.
**** ∏ AC
LM MK
2.
4x x 15
10.
x 10
12.
HE HF
3x x 4
MK x 15 5 15 20
3x x x 4 x
EF GF
2x 4
2x
4
2
x2
2x 5 x 10
2x 5 x x 10 x
x 5 10
13. BC
**** BD
**** by the Angle Bisector Theorem.
x 5 5 10 5
14. ma JKM ma LKM 38
x5
15. SV UV 18
EF 2x 5 2(5) 5 10 5 15
17.
5.6 Guided Practice (p. 276)
LK NK
x x 2x 3 x
equidistant from the two sides of the angle.
0x3
2. If D is on the perpendicular bisector of ****
AB , then D is
03x33
equidistant from A and B.
3x
3. AD DC 16
LK x 3
EF FG
18.
x 1 2x 1
PQ RQ
x 2 2x 1
x 1 x 2x 1 x
x 2 x 2x 1 x
1x1
2x1
11x11
21x11
2x
3x
EF x 1 2 1 3
PQ x 2 3 2 5
5. JM ML 12
19. AD CD 12;
QR QP
3x 8 5x
BC BA
3x 8 3x 5x 3x
2x 6 3x 1
2x 6 2x 3x 1 2x
8 2x
8
2x
2
2
4x
6x1
61x11
5x
QR 3x 8 3(4) 8 12 8 20
BC 2x 6 2(5) 6 10 6 16
20. Perpendicular Bisector Theorem; Because DB
**** is perpen-
5.6 Practice and Applications (pp. 276–280)
7.
dicular to and bisects ****
AC , point D is on the perpendicular
bisector of ****
AC . So, D is equidistant from points A and C.
8.
A
C
A
B
D
C
B
D
78
16. HG FG 14
x 2x 3
1. If a point is on the bisector of an angle, then it is
6.
x6
x 5 5 15 5
3x 15
3x
15
3
3
x5
4.
11. MJ ML
x 5 15
4x x x 15 x
3.
CB CA
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
21. Line l is the perpendicular bisector of the goal line (AC
**** ).
22. BG
&*( should bisect a ABC so that the goalkeeper is
equidistant from the edges of the region that the goalkeeper is defending. By doing this, the goalkeeper
minimizes the distance needed to reach the ball.
Copyright © McDougal Littell Inc.
All rights reserved.
Chapter 5 continued
23. CG AG BG 2
24. VR VQ VP 16
25. GL
**** LH
**** ;
5.6 Standardized Test Practice (p. 280)
34. B; SR SP
FJ
JG ; FK
**** ****
**** KH
**** ;
x 2x 3
FM
GM HM
**** **&
*&
***
26 –28.
x x 2x 3 x
0x3
A
03x33
(red)
3x
X
(blue)
(gray)
SR x 3
B
35. G;
C
29. The fire station at point A; X is in the red region, which
2x 3 x x x
x30
is the region closest to station A.
30. DA
**** DB
**** ; the measures of DA
**** and DB
**** are equal.
x3303
x3
31. Yes; by the Perpendicular Bisector Theorem, if D lies on
the perpendicular bisector of ****
AB , then DA
**** and DB
**** will
always be congruent segments.
32. Statements
SP RS
2x 3 x
Reasons
SP 2x 3 2(3) 3 6 3 3
36. D
1. ^&(
AD is the perpendicular
bisector of BC
**** .
1. Given
2. DB
**** DC
****
2. Definition of perpendicular bisector
3. a ADB and a ADC
are right angles.
3. Perpendicular lines
form right angles.
4. a ADB a ADC
4. Right angles are .
41. (6, 2) → (6 3, 2 6) (9, 4)
5. AD
**** AD
****
5. Reflexive Prop. of Cong.
42. (2, 5) → (2 3, 5 6) (5, 11)
6. T ADB T ADC
6. SAS Cong. Postulate
43. (10, 12) → (10 3, 12 6) (13, 6)
7. ****
AB ****
AC
7. Corresponding parts of
triangles are .
44. (1, 1) → (1 3, 1 6) (2, 7)
8. AB AC
8. Def. of segments
33. Statements
Reasons
5.6 Mixed Review (p. 280)
37. (5, 1) → (5 3, 1 6) (8, 5)
38. (2, 3) → (2 3, 3 6) (1, 3)
39. (4, 4) → (4 3, 4 6) (1, 10)
40. (0, 6) → (0 3, 6 6) (3, 12)
45. SAS Congruence Postulate; DB
**** DB
**** (Reflexive
Property of Congruence) and all right angles are
congruent, so two sides and the included angle of
T ADB are congruent to two sides and the included
angle of T CDB.
1. D is on the bisector
of a BAC.
1. Given
2. a BAD a CAD
2. Definition of angle
bisector
3. DB
&*( and
**** ∏ AB
DC
AC
*(
**** ∏ &
3. Given
4. a DBA and a DCA
are right angles.
4. Perpendicular lines
form right angles.
5. a DBA a DCA
5. Right angles are .
5.6 Algebra Skills (p. 280)
6. DA
**** DA
****
6. Reflexive Prop. of Cong.
7. T ADB T ADC
48. 3, 0.3, 0, 0.3, 0.6, 3
7. AAS Cong. Theorem
8. DB
**** DC
****
49. 1.25, 0.75, 0.25, 0.25, 1, 4
8. Corresponding parts of
triangles are .
50. 4, 0.4, 0.1, 0, 0.1, 4.0
Copyright © McDougal Littell Inc.
All rights reserved.
46. AAS Congruence Theorem; EF
**** EF
**** (Reflexive
Property of Congruence), so two angles and a nonincluded side of T HEF are congruent to two angles
and the corresponding non-included side of T GFE.
47. ASA Congruence Postulate; vertical angles are congruent
so two angles and the included side of T JNK are congruent to two angles and the included side of T LNM.
51. 3.9, 3.3, 3, 3.1, 3.5, 3.8
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
79
Chapter 5 continued
52. 1, 0.1, 0, 0.5, 0.55, 1, 1.1
62.
53. 2.5, 1, 2.1, 3.2, 3.25, 5
4x 3 11
54.
10t 10 10t 12t 10t
55.
4x 3 3 11 3
4x 8
4x
8
4
4
x2
2y 9 11
2y 2
2y
2
2
2
y 1
5d 35 35 90 35
5d 125
5d
125
5
5
d 25
57. 4a 9a 39
13a 39
13a
39
13
13
a3
x 2 3x 4
58.
10 2t
10
2t
2
2
5t
2y 9 9 11 9
5d 35 90
56.
10t 10 12t
x 2 x 3x 4 x
Lesson 5.7
5.7 Activity (p. 281)
XA **&
AX, ZB
1. Answers will vary, but ****
**** BZ
**&, and
YC
*& CY
**&.
XA **&
AX, ZB
2. ****
**** BZ
**&, and YC
*& CY
**&; each pair of
segments is congruent.
3. 90; 90; 90; the measure of each angle is 90.
4. The fold line is the perpendicular bisector of each
segment.
5.7 Checkpoint (pp. 283–285)
1. yes; the x-axis
2. no
4. The figure has one
line of symmetry.
3. yes; the y-axis
5. The figure has two
lines of symmetry.
2 2x 4
2 4 2x 4 4
6 2x
6
2x
2
2
3x
6. The figure has four lines of symmetry.
4r 2 5r 6
59.
4r 2 4r 5r 6 4r
2 r 6
2 6 r 6 6
8 r
q 2q 9
60.
q 2q 2q 9 2q
q 9
q
9
1
1
q9
2z 5 4z 1
61.
2z 5 2z 4z 1 2z
5.7 Guided Practice (p. 286)
1. A figure in the plane has a line of symmetry if the figure
can be reflected onto itself by a reflection in the line.
2. No; the orientation of the image is not reversed.
3. Yes; all three properties of a reflection are met.
4. Yes; all three properties of a reflection are met.
5. The flower has three
lines of symmetry.
6. The flower has two
lines of symmetry.
5 2z 1
5 1 2z 1 1
6 2z
6
2z
2
2
3z
80
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
Copyright © McDougal Littell Inc.
All rights reserved.
Chapter 5 continued
7. The flower has five lines of symmetry.
22. yes
23. yes
24. The figure has five
25. The figure has four
lines of symmetry.
lines of symmetry.
26. The figure has two lines of symmetry.
5.7 Practice and Applications (pp. 286–290)
8. Yes; all three properties of a reflection are met.
9. Yes; all three properties of a reflection are met.
10. No; the orientation is not reversed.
27. All the lines of symmetry are shown.
11. The grid shows a reflection in the y-axis.
28. no;
29. no;
12. The grid shows neither a reflection in the x-axis, nor a
reflection in the y-axis.
13. The grid shows a reflection in the x-axis.
14. GH
AB in the x-axis; point G
**** is a reflection of ****
corresponds to A; point H corresponds to B.
AB in the y-axis; point C
15. CD
**** is a reflection of ****
corresponds to A; point D corresponds to B.
16. Coordinates of ****
AB : A(3, 1), B(1, 3).
Coordinates of GH
**** : G(3, 1), H(1, 3).
The coordinates are alike in that the x-coordinate of each
original point is the same as the x-coordinate of the
image point. The coordinates are different in that the
y-coordinate of each original point is the opposite of the
y-coordinate of the image point.
17.
18.
30.
31. The letters b and d are reflections of each other, and p
and q are reflections of each other.
k
32. One line of symmetry:
,
,
, and
;
k
Two lines of symmetry:
19.
,
,
33. yes;
k
34. no
35. yes;
36. yes;
20. All three triangles in sections B, and D are reflections of
the triangle in section A. Each fold in the paper is a line
of symmetry.
21. No; the right side of the guitar is not the same as the
left side.
Copyright © McDougal Littell Inc.
All rights reserved.
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
81
Chapter 5 continued
180
n
180
4
45
180
n
180
2
90
37. ma A 38. ma A 180
n
180
3
60
48. ma 1 51 30 180
ma 1 81 180
ma 1 81 81 180 81
ma 1 99
ma 1 44 90 180
49.
39. ma A ma 1 134 180
ma 1 134 134 180 134
ma 1 46
40. AB ⏐3 1⏐ 2, DE ⏐3 (1)⏐ ⏐2⏐ 2;
AC ⏐6 2⏐ 4, DF ⏐6 2⏐ 4;
2
)2
(1
3
)2 4
2(
2
)2 2
0
,
BC (6
EF (6
2
)2
(
1
(
3))
2 4222 20;
By the SSS Congruence Postulate, T ABC T DEF.
41. AB ⏐1 3⏐ ⏐2⏐ 2,
DE ⏐1 (3)⏐ ⏐2⏐ 2;
AC ⏐4 1⏐ 3,
DF ⏐4 (1)⏐ ⏐3⏐ 3;
5.7 Algebra Skills
50. 2348 2384
51. 5 7
52. 19.1 19.01
53. 11.2 11.238
54. 0.065 0.056
55. 1.011 1.11
Quiz 3 (p. 290)
1. A
1
)2
(4
1
)2 2
232 13,
BC (3
B
EF (
4
(
1))
2(
3
(
1))
2
(
3
)2
(
2
)2 1
3
;
By the SSS Congruence Postulate, T ABC T DEF.
C
B
C
AAS Congruence Postulate
2. DC AD 2
3. ML MJ 9;
JK LK 25
5.7 Standardized Test Practice (p. 289)
4.
42. C
D
AB CD
3x 4 2x 7
43. G;
3x 4 2x 2x 7 2x
x47
x4474
5.7 Mixed Review (p. 290)
44. x 105
x3
AB 3x 4 3(3) 4 13
5. The figure has one line of symmetry.
x 105
(x 10) 82
45.
x 10 10 82 10
x 72
(3x 1) 92
46.
3x 1 1 92 1
6. The figure has two lines
of symmetry.
7. The figure has three
lines of symmetry.
3x 93
3x
93
3
3
x 31
ma 1 38 75 180
47.
ma 1 113 180
Chapter 5 Summary and Review (pp. 291–295)
ma 1 113 113 180 113
1. When two figures are congruent, their corresponding
ma 1 67
82
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
sides and their corresponding angles are congruent.
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Chapter 5 continued
2. A proof is a convincing argument that shows why a
23. Statements
statement is true.
1. a JMN and a JKL
are right angles.
3. If a point is on the perpendicular bisector of a segment,
then it is equidistant from the endpoints of the segment.
4. If a point is on the angle bisector of an angle, then it is
equidistant from the two sides of the angle.
5. A reflection is a transformation that creates a mirror
image.
6. neither
7. corresponding sides
8. corresponding angles
9. neither
10. neither
11. corresponding angles
because it is given that DE
AC , a E a C, and
**** ****
EF
**** CB
**** .
13. Yes; T PTQ T RTS by the SAS Congruence Postulate
because PT
**** RT
**** (Given), a PTQ a RTS (Vertical
Angles Theorem), and ST
**** QT
**** (Given).
14. Yes; T PQR T PSR by the SSS Congruence Postulate
15. a S a X
16. ****
AB FG
****
18. Statements
19.
17. a JKM a LMK
2. a JMN a JKL
2. All right angles are .
3. a J a J
3. Reflexive Prop. of Cong.
4. MN
**& KL
****
5. T JMN T JKL
4. Given
6. JM
**** JK
****
6. Corresponding parts of triangles are .
Reasons
1. Given
2. a Q a S
2. All right angles are .
3. a QRP a SRT
3. Vertical Angles Theorem
4. ****
PR TR
*&
5. T QRP T SRT
4. Given
6. QR
**** SR
****
6. Corresponding parts of triangles are .
26.
QR QP
2x 1 x 4
1. Given
2x 1 x x 4 x
2. T UZV and T XYW
are right triangles.
2. Definition of right triangle
3. UV
XW
**** **&
4. UZ
**** XY
****
5. T UZV T XYW
3. Given
QR 2x 1 2(5) 1 9
4. Given
5. HL Congruence Theorem
A
B
L
M
J
J
K
N
AAS Congruence Theorem
21. P
S
R
R
T
AAS Congruence Theorem
22. Statements
XY ZY
3x 4 6x 1
3x 4 3x 6x 1 3x
4 1 3x 1 1
3 3x
3
3x
3
3
1x
XY 3x 4 3(1) 4 7
28. Yes; all three properties of a reflection are met; one line
Reasons
1. a C and a D
are right angles.
1. Given
2. T ABC and T BAD
are right triangles.
2. Definition of right triangle
3. ****
AC BD
****
4. ****
AB ****
AB
3. Given
of symmetry.
29. No; the orientation is not reversed; one line of symmetry.
30. Yes; all three properties of a reflection are met; no lines
of symmetry.
31. Sample answer:
4. Reflexive Prop. of Cong.
5. T ABC T BAD
5. HL Congruence Theorem
6. a CBA a DAB
6. Corresponding parts of triangles are .
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27.
4 3x 1
HL Congruence Theorem
20.
x14
x1141
x5
D
B
5. AAS Congruence Theorem
25. JM LM 3
1. a UZV and a XYW
are right angles.
A
5. AAS Congruence Theorem
1. a Q and a S are
right angles.
Reasons
C
1. Given
24. Statements
12. Yes; T DEF T ACB by the SAS Congruence Postulate
because PQ
**** PS
**** (Given), QR
**** SR
**** (Given), and
PR ****
PR (Reflexive Property of Congruence).
****
Reasons
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
83
Chapter 5 continued
Chapter 5 Chapter Test (p. 296)
12. The figure has two
13. The figure has four
lines of symmetry.
1. RS
**** XY
**** , ST
**** YZ
**** , TR
*& ZX
**** ,
lines of symmetry.
a R a X, a S a Y, a T a Z;
T RST T XYZ, T STR T YZX,
T TRS T ZXY, T TSR T ZYX,
T SRT T YXZ, or T RTS T XZY
2. ma L ma U 46
3. TU KL 10
4. SSS Congruence Postulate; BC
**** BC
**** (Reflexive
Property of Congruence), so all three sides of T ABC are
congruent to all three sides of T DBC.
Chapter 5 Standardized Test (p. 297)
1. C
2. J
3. B
AD CD
4. F;
4x 3 2x 3
5. HL Congruence Theorem; by the Reflexive Property of
Congruence, HF
**** HF
****, so the hypotenuse and leg of
right T HFE are congruent to the hypotenuse and corresponding leg of T HFG.
4x 3 2x 2x 3 2x
2x 3 3
2x 3 3 3 3
6. Yes; vertical angles are congruent, so the triangles are
2x 6
2x
6
2
2
x3
congruent by the AAS Congruence Theorem.
7. Yes; vertical angles and alternate interior angles are
congruent, so the triangles are congruent by the AAS
Congruence Theorem.
Reasons
5. C; AD 4x 3 4(3) 3 12 3 9
1. BC
**** DC
****
1. Given
6. J;
2. a B and a D are
right angles.
2. Given
3. a B a D
3. Right angles are .
4. a ACB a ECD
4. Vertical Angles Theorem
5. T ABC T EDC
5. ASA Cong. Postulate
AB ED
6. ****
****
6. Corresponding parts of triangles are .
8. Statements
7.
y
B
A
1
9. PR PQ
C
1
x
Cⴕ
2x 8
2x
8
2
2
x4
8. A is (2, 2); B is (3, 4); C is (5, 1).
PR 2x 2(4) 8
9. The x-coordinate of each vertex in the original triangle is
Aⴕ
Bⴕ
the same as the x-coordinate of the vertex in the reflected
triangle. The y-coordinate of each vertex in the original
triangle is the opposite of the y-coordinate of the vertex
in the reflected triangle.
ST UT
10.
5x 2x 3
5x 2x 2x 3 2x
3x 3
3x
3
3
3
x1
ST 5x 5(1) 5
11. No; the orientation of the figure is not reversed.
Chapter 5 Algebra Review (p. 299)
1.
3x 2y 13
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
5x 4y 4
3(3) 2(2) ⱨ 13
5(0) 4(1) ⱨ 4
9 4 ⱨ 13
0 (4) ⱨ 4
13 13
44
(3, 2) lies on the line.
84
2.
(0, 1) lies on the line.
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All rights reserved.
Chapter 5 continued
3. y 3x 8
4.
4 ⱨ 3(4) 8
x 6y 2
(4) 6(1) ⱨ 2
4 ⱨ 12 8
4 6 ⱨ 2
4 20
2 2
(4, 4) does not lie
on the line.
5.
x 7y 5
(4, 1) does not lie
on the line.
2x 4y 2
6.
(5) 7(0) ⱨ 5
2(1) 4(1) ⱨ 2
5 0 ⱨ 5
2 4 ⱨ 2
5 5
2 2
(5, 0) does not lie
on the line.
(1, 1) does not lie
on the line.
y2 y1
71
6
7. slope x2 x1
83
5
y2 y1
5 4
9
8. slope not defined
x2 x1
33
0
y2 y1
7
1 (6)
9. slope 1
x2 x1
7
1 6
y2 y1
7
81
10. slope x2 x1
3
2 (5)
y2 y1
44
0
11. slope 0
x2 x1
4 (1)
5
y2 y1
6 (4)
10
5
12. slope x2 x1
2 (2)
4
2
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All rights reserved.
Geometry, Concepts and Skills
Chapter 5 Worked-Out Solution Key
85