close

Вход

Log in using OpenID

embedDownload
Lesson 6.1 Skills Practice
Name
Date
Time to Get Right
Right Triangle Congruence Theorems
Vocabulary
Choose the diagram that models each right triangle congruence theorem.
1. Hypotenuse-Leg (HL) Congruence Theorem
a. X
Y
R
Q
P
Z
2. Leg-Leg (LL) Congruence Theorem
b. U
V
W
X
3. Hypotenuse-Angle (HA) Congruence Theorem c. E
F
© Carnegie Learning
G
I
4. Leg-Angle (LA) Congruence Theorem
H
d. 6
W
T
U
X
V
Y
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 587
587
17/06/13 11:37 AM
Lesson 6.1 Skills Practice
page 2
Problem Set
Mark the appropriate sides to make each congruence statement true by the Hypotenuse‑Leg
Congruence Theorem.
1. DPR  QFM
R
2.ACI  GCE
A
M
I
C
F
D
E
Q
P
G
3. QTR  SRT
T
4.ADG  HKN
Q
D
A
S
N
K
R
H
G
Mark the appropriate sides to make each congruence statement true by the Leg‑Leg
Congruence Theorem.
5. BZN  TGC
6.MNO  QPO
G
T
N
P
B
O
6
Z
C
7. PZT  PZX
M
Q
8.EGI  ONQ
P
E
G
N
I
Q
O
© Carnegie Learning
N
T
Z
X
588 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 588
17/06/13 11:37 AM
Lesson 6.1 Skills Practice
page 3
Name
Date
Mark the appropriate sides and angles to make each congruence statement true by the
Hypotenuse‑Angle Congruence Theorem.
9. SVM  JFW
10.MSN  QRT
W
F
V
P
J
S
N
T
M
S
11. IEG  IEK
M
Q
R
12.DCB  ZYX
E
G
Y
D
B
I
X
K
C
Z
Mark the appropriate sides and angles to make each congruence statement true by the Leg‑Angle
Congruence Theorem.
13. XTD  HPR
14.SEC  PEC
P
X
T
S
R
© Carnegie Learning
T
C
D
D
E
6
H
P
R
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 589
589
17/06/13 11:37 AM
Lesson 6.1 Skills Practice
15. PBJ  OTN
page 4
16.AXT  YBU
P
A
X
B
Y
B
J
T
O
T
U
N
For each figure, determine if there is enough information to prove that the two triangles are congruent.
If so, name the congruence theorem used.
___
___
___
17. Given: GF 
​ ​ bisects /RGS, and 18. Given: DV 
​ ​  TU 
​ ​ 
/R and /S are right angles. Is nDVT > nDVU?
Is nFRG > nFSG?
G
D
S
R
T
V
U
F
es. There is enough information
Y
to conclude that nFRG > nFSG
by HA.
____
____ ____
____
___
___
19. Given: NM 
​ ​ > EM 
​ ​, DM 
​ ​ > OM 
​ ​, and
20. Given: RP 
​ ​ > QS 
​ ​, and /R and /Q
/NMD and /EMO are right angles. are right angles.
Is nNMD > nEMO? Is nSRP > nPQS?
N
O
6
M
S
P
Q
D
E
590 R
© Carnegie Learning
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 590
17/06/13 11:37 AM
Lesson 6.1 Skills Practice
page 5
Name
Date
____
____
___
____
21. Given: GO 
​ ​ > MI 
​ ​, and /E and /K are
22. Given: HM 
​ ​ > VM 
​ ​, and /H and /V are
right angles. right angles.
Is nGEO > nMKI? Is nGHM > nUVM?
E
G
I
G
H
M
O
M
K
V
U
Use the given information to answer each question.
23. T
wo friends are meeting at the library. Maria leaves her house and walks north on Elm Street and then
east on Main Street to reach the library. Paula leaves her house and walks south on Park Avenue and
then west on Main Street to reach the library. Maria walks the same distance on Elm Street as Paula
walks on Main Street, and she walks the same distance on Main Street as Paula walks on Park
Avenue. Is there enough information to determine whether Maria’s walking distance is the same as
Paula’s walking distance?
Paula’s house
W
Park Avenue
N
E
S
Elm Street
© Carnegie Learning
Main Street
Library
Maria’s house
Yes. Maria’s walking distance to the library is equal to Paula’s walking distance. The triangles
formed are right triangles. The corresponding legs of the triangles are congruent. So, by the
Leg-Leg Congruence Theorem, the triangles are congruent. If the triangles are congruent, the
hypotenuses are congruent.
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 591
6
591
17/06/13 11:37 AM
Lesson 6.1 Skills Practice
page 6
24. A
n auto dealership displays one of their cars by driving it up a ramp onto a display platform. Later
they will drive the car off the platform using a ramp on the opposite side. Both ramps form a right
triangle with the ground and the platform. Is there enough information to determine whether the two
ramps have the same length? Explain.
Display
platform
Ground
25. A
radio station erected a new transmission antenna to provide its listeners with better reception. The
antenna was built perpendicular to the ground, and to keep the antenna from swaying in the wind
two guy wires were attached from it to the ground on opposite sides of the antenna. Is there enough
information to determine if the guy wires have the same length? Explain.
guy wire
Tower
guy wire
© Carnegie Learning
Ground
6
592 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 592
17/06/13 11:37 AM
Lesson 6.1 Skills Practice
page 7
Name
Date
26. T
wo ladders resting on level ground are leaning against the side of a house. The bottom of each
ladder is exactly 2.5 feet directly out from the base of the house. The point at which each ladder rests
against the house is 10 feet directly above the base of the house. Is there enough information to
determine whether the two ladders have the same length? Explain.
Create a two-column proof to prove each statement.
____
___ ____
___
___
___
27. Given: WZ 
​ ​ bisects VY 
​ ​, WV 
​ ​  VY 
​ ​, and YZ 
​ ​  VY 
​ ​
 
W
Prove: WVX  ZYX
X
Y
V
© Carnegie Learning
Z
____ ___ Statements
___ ___
1. ​WV ​  VY ​
​  and YZ 
​ ​  VY ​
​  
Reasons
1. Given
2. WVX and ZYX are right angles.
2. Definition of perpendicular angles
3. WVX and ZYX are right triangles.
____
___
4. ​WZ ​ bisects VY 
___ ___ ​ ​. 
5. ​VX ​  YX ​
​  
3. Definition of right triangles
6. WXV  ZXY
6. Vertical Angle Theorem
7. WVX  ZYX
7. LA Congruence Theorem
4. Given
5. Definition of segment bisector
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 593
6
593
17/06/13 11:37 AM
Lesson 6.1 Skills Practice
page 8
___
28. G
iven: Point D is the midpoint of EC 
​ ​,  A
ADB
is an isosceles triangle with base
___
AB 
​ ​, and E and C are right angles.
Prove: AED  BCD
___
___ ___
E
___
___
___
​ ​, TP 
​ ​  UP 
​ ​, and UR 
​ ​  PR 
​  ​ 29. Given: SU 
​ ​  UP 
B
D
T
C
P
Prove: SUR  TPR
R
6
594 U
S
© Carnegie Learning
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 594
17/06/13 11:37 AM
Lesson 6.1 Skills Practice
page 9
Name
Date
30. Given: Rectangle MNWX and NMW  XWM M
Prove: MNW  WXM
X
W
© Carnegie Learning
N
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 595
595
17/06/13 11:37 AM
© Carnegie Learning
6
596 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 596
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
Name
Date
CPCTC
Corresponding Parts of Congruent Triangles are Congruent
Vocabulary
Provide an example to illustrate each term. 1. Corresponding parts of congruent triangles are congruent (CPCTC)
2. Isosceles Triangle Base Angle Theorem
© Carnegie Learning
3. Isosceles Triangle Base Angle Converse Theorem
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 597
597
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
page 2
Problem Set
Create a two-column proof to prove each statement.
___
___
1. Given: RS 
​ ​ is the  bisector of PQ 
​ ​. 
Q
Prove: SPT  SQT
T
R
P
Statements
___
___
1. ​RS ​ is the  bisector of PQ ​
​ . 
___ ___
2. ​RS ​  PQ ​
​  
Reasons
1. Given
3. PTS and QTS are right angles.
3. Definition of perpendicular lines
4. PTS
___ and QTS
___ are right triangles.
4. Definition of right triangles
2. Definition of perpendicular bisector
5. ​RS ​ bisects PQ ​
​  
___ ___
___ ___
7. ​TS ​  TS 
​ ​ 
5. Definition of perpendicular bisector
8. PTS  QTS
8. Leg-Leg Congruence Theorem
9. SPT  SQT
9. CPCTC
6. ​PT ​  QT 
​ ​ 
___
____ ___
6. Definition of bisect
7. Reflexive Property of 
____
___
____
​ ​, TM 
​ ​  WT 
​ ​, and TZ 
​ ​  WX 
​ ​ 
2. Given: TZ 
​ ​  WX 
M
____
___
Prove: MZ 
​ ​  TX 
​ ​ 
T
Z
X
6
598 W
© Carnegie Learning
S
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 598
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
page 3
Name
Date
___
___
3. G
___
iven:___
AG 
​ ​___
 
and EK 
​ ____
​ intersect at C, AC 
​ ​  EC 
​ ​, CK 
​ ​  CG 
​ ​ 
A
E
C
Prove: K  G
K
G
4. Given: JHK  LHK, JKH  LKH ___
H
___
Prove: JK 
​ ​  LK 
​ ​ 
L
J
K
© Carnegie Learning
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 599
599
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
page 4
5. Given: UGT  SGB B
T
Prove: TUS  BSU
G
S
U
___
___
6. Given: TPN  TNP, ​TP  ​ QP ​
​  
P
Prove:
___
___
TN ​
​   QP ​
​  
T
6
600 N
© Carnegie Learning
Q
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 600
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
page 5
Name
Date
___
___ ___
___
7. Given: AC 
​ ​  DB 
​ ​, AC 
​ ​ bisects DB 
​ ​  
A
___
___
Prove: AD 
​ ​  AB 
​ ​ 
E
D
C
___
___ ___
___
​ ​, FK 
​ ​  JK 
​ ​  
8. Given: KGH  KHG, FG 
​ ​  JH 
© Carnegie Learning
G
H
Prove: F  J
F
B
K
J
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 601
601
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
___
page 6
___ ___
​ ​, AC 
​ ​ bisects TAQ
9. Given: AT 
​ ​  AQ 
T
___
___
Prove: AC 
​ ​ bisects TQ 
​ ​ 
C
Q
A
___
__
__
​ ​,  LNJ  IGJ, J is the midpoint of LI 
​ ​  10. Given: EL 
​ ​  EI 
E
___
___
Prove: NJ 
​ ​  GJ 
​ ​ 
N
L
I
6
602 J
© Carnegie Learning
G
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 602
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
page 7
Name
Date
11. Given: E  EUV, F  FVU E
___
___
Prove: UF 
​ ​  VE 
​ ​ 
U
V
F
___
___ ___
___
12. Given: CT 
​ ​  CP 
​ ​, AT 
​ ​  AP 
​ ​  
C
Prove: mCTA 5 mCPA
A
T
© Carnegie Learning
P
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 603
603
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
page 8
Use the given information to answer each question.
13. S
amantha is hiking through the forest and she comes upon a canyon. She wants to know how wide
the canyon is. She measures the distance between points A and B to be 35 feet. Then, she measures
the distance between points B and C to be 35 feet. Finally, she measures the distance between
points C and D to be 80 feet. How wide is the canyon? Explain.
A
D
B
E
C
The canyon is 80 feet wide.
he triangles are congruent by the Leg-Angle
T
____ ___Congruence Theorem. Corresponding parts of
congruent triangles are congruent, so ​CD ​ 5 ​AE ​. 
14. Explain why mNMO 5 20°.
M
60°
Q
6
80°
P
O
N
15. Calculate MR given that the perimeter of HMR is 60 centimeters.
M
60°
20 cm
© Carnegie Learning
60°
H
604 R
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 604
17/06/13 11:37 AM
Lesson 6.2 Skills Practice
Name
page 9
Date
16. Greta has a summer home on Lake Winnie. Using the diagram, how wide is Lake Winnie?
Lake
Winnie
52 m
20 m
20 m
Greta’s
summer home
52 m
48 m
17. J
ill is building a livestock pen in the shape of a triangle. She is using one side of a barn for one of the
sides of her pen and has already placed posts in the ground at points A, B, and C, as shown in the
diagram. If she places fence posts every 10 feet, how many more posts does she need? Note: There
will be no other posts placed along the barn wall.
A
Barn wall
Livestock
pen
B
50'
© Carnegie Learning
C
6
18. Given rectangle ACDE, calculate the measure of CDB.
A
B
C
30°
E
D
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 605
605
17/06/13 11:37 AM
© Carnegie Learning
6
606 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 606
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
Name
Date
Congruence Theorems in Action
Isosceles Triangle Theorems
Vocabulary
Choose the term from the box that best completes each sentence.
Isosceles Triangle Altitude to Congruent
Sides Theorem
Isosceles Triangle Base Theorem
Isosceles Triangle Vertex Angle Theorem
Isosceles Triangle Angle Bisector to Congruent
Sides Theorem
Isosceles Triangle Perpendicular Bisector
Theorem
1. A
(n)
two congruent legs in an isosceles triangle.
vertex angle
is the angle formed by the
2. In an isosceles triangle, the altitudes to the congruent sides are congruent,
as stated in the
.
3. In an isosceles triangle, the angle bisectors to the congruent sides are congruent, as stated in the
.
© Carnegie Learning
4. T
he
states that the altitude
from the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
5. T
he
the base of an isosceles triangle bisects the base.
states that the altitude to
6. The altitude to the base of an isosceles triangle bisects the vertex angle, as stated in the
.
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 607
6
607
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
page 2
Problem Set
Write the theorem that justifies the truth of each statement.
___
____
___
1. In isosceles MRG, RD 
​ ​  GC 
​ ​. 
2.___
In isosceles
TGC
with
altitude TP ​
​ , 
____
___
___
TP 
​ ​  GC 
​ ​, and GP 
​ ​  CP 
​ ​. 
G
R
C
T
G
D
P
M
C
Isosceles Triangle Angle Bisector to Congruent Sides Theorem
___
___
3. In
isosceles
BRU with altitude BD 
​ ​, 
4. In isosceles JFI with altitude JH 
​ ​, 
___
___
UD 
​ ​  RD 
​ ​.  /HJF  /HJI.
B
I
H
R
F
D
U
J
___
___
O K
N
M
6
B
A
H
M
© Carnegie Learning
5. In isosceles nMNO, OA 
​ ​ > NB.
6. In
isosceles nHJK, KN 
​ ​ ___
bisects
/HKJ,
___
___
​JM ​ bisects /HJK, and MJ 
​ ​ > NK 
​ ​. 
N
J
608 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 608
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
page 3
Name
Date
Determine the value of x in each isosceles triangle.
7.
B
8 in.
S
25 m
32 °
x°
D
8.
K
24 m
8 in.
A
25 m
P
C
x
J
W
x 5 32°
9.
10.
M
20 ft
P
x
S
4m V
20 ft
D
26 ft
x
U
N
16 m
16 m
20°
20°
T
© Carnegie Learning
11.
W
12.
P
Q
10 cm
V
29 yd
x
12 cm
12 cm
37°
x
R
29 yd
T
U
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 609
6
609
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
page 4
Complete each two-column proof.
___
___
AB 
​  CB 
​ ​,  13. G
___
iven:___
Isosceles
ABC with
​ ___
___
___
___
BD 
​ ​  AC 
​ ​, DE 
​ ​  AB 
​ ​, and DF 
​ ​  CB 
​ ​ 
___
B
___
E
Prove: ED 
​ ​  FD 
​ ​ 
A
D
___ ____ Statements
1. ​AB ​  CB ​
​  
___ ___ ____
___ ___
____
2. ​BD ​  AC ​
​ , DE ​
​   AB ​
​ , DF ​
​   CB 
​ ​ 
2. Given
3. AED and CFD are right angles.
3. Definition of perpendicular lines
4. AED and CFD are right triangles.
4. Definition of right triangle
5. A  C
5. Base Angle Theorem
C
Reasons
1. Given
6. Isosceles Triangle Base Theorem
7. HA Congruence Theorem
8. CPCTC
© Carnegie Learning
___ ____
6. ​AD ​  CD ​
​  
7. AED  CFD
___ ___
8. ​ED ​  FD 
​ ​ 
6
610 F
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 610
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
page 5
Name
Date
____
____
14.Given:
Isosceles MNB
with MN 
​ ​  MB 
​ ​, 
____
___
NO 
​ ​ bisects ANB, BA 
​ ​ bisects OBN
M
Prove: BAN  NOB
B
A
N
© Carnegie Learning
O
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 611
611
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
__
__ ___
page 6
__ ___
__
15. Given: Isosceles IAE with IA 
​ ​  I​E ​,  AG 
​ ​  IE 
​ ​, EK 
​ ​  IA 
​ ​ 
A
E
Prove: IGA  IKE
M
K
G
I
___
____
​____​, 
16. Given: Isosceles GQR with GR 
​____​  GQ 
Isosceles
QGH____
with GQ 
​ ___
​  QH 
​ ___
​, 
___
___ ___
GJ 
​ ​  QR 
​ ​, QP 
​ ​  GH 
​ ​, and GJ 
​ ​  QP 
​ ​ 
___
G
P
J
Q
H
___
Prove: RJ 
​ ​  HP 
​ ​ 
R
© Carnegie Learning
6
612 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 612
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
page 7
Name
Date
Use the given information to answer each question.
17. T
he front ___
of an A-frame
house
is in the shape of an isosceles triangle, as shown in the diagram. In the
___ ____
___
diagram, HK 
​ ​  GJ 
​ ​, GH 
​ ​  JH 
​ ​, and mHGJ 5 68.5°. Use this information to determine the measure of
GHJ. Explain.
H
G
K
J
The measure of GHJ is 43°.
By the Triangle Sum Theorem, mGHK 5 180° 2 (90° 1 68.5°) 5 21.5°.
y the Isosceles Triangle Vertex Angle Theorem, mGHK 5 mJHK.
B
Therefore, mGHJ 5 21.5° 1 21.5° 5 43°.
18. W
hen building a house, rafters are used to support the___
roof. The
rafter shown
in the diagram
___ ___
____
has the shape of an isosceles triangle. In the diagram, NP 
​ ​  RQ 
​ ____​, NR 
​ ​  NQ 
​ ​, NP 5 12 feet, and
RP 5 16 feet. Use this information to determine the length of NQ 
​ ​. Explain.
© Carnegie Learning
N
R
P
Q
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 613
613
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
page 8
19. S
tained glass windows are constructed using different pieces of colored glass held together by lead.
The stained glass window in the diagram is rectangular with six different colored glass pieces___
represented by TBS,
PBS, PBQ, QBR, NBR, and NBT. Triangle TBP with altitude SB 
​ ​ and
___
QBN with altitude RB 
​ ​, are congruent isosceles triangles. If the measure of NBR is 20°, what is the
measure of STB? Explain.
T
S
P
B
N
R
Q
20. W
hile growing up, Nikki often camped out in her back yard in a pup tent. A pup tent has two
rectangular sides made of canvas, and a front and back
in the shape of two isosceles triangles also
____
made of canvas. The zipper in front, represented by ___
MG 
​ ​ in the diagram, is the height of ___
the pup tent
and the altitude of isosceles EMH. If the length of EG 
​ ​ is 2.5 feet, what is the length of HG 
​ ​?  Explain.
K
M
6
614 E
G
T
H
© Carnegie Learning
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 614
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
page 9
Name
Date
___
___ ____
___
21. A
beaded
purse
is in the shape of an isosceles triangle. In the
diagram, TN 
​ ​  TV 
​ ​, VM 
​ ​  TN 
​ ​, 
___
___
___
and NU 
​ ​  TV 
​ ​. How long is the line of beads represented by NU 
​ ​, if TV is 13 inches and TM is
5 inches? Explain.
T
M
R
V
© Carnegie Learning
N
U
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 615
615
17/06/13 11:37 AM
Lesson 6.3 Skills Practice
page 10
22. A
kaleidoscope is a cylinder with mirrors inside and an assortment of loose colored beads.
When a person looks through the kaleidoscope, different colored shapes and patterns are
created as the kaleidoscope is rotated. Suppose that the diagram represents the shapes that a
person
sees when they look into
the kaleidoscope. Triangle AEI is an isosceles
triangle with
___
___
___
__ ___
AE 
​ . EK 
​ ​ bisects AEI and IC 
​ ​ bisects AIE. What is the length of IC, 
​  ​if one half the length of ​
​___​  AI ​
EK ​ is 14 centimeters? Explain.
A
K
C
M
E
© Carnegie Learning
I
6
616 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 616
17/06/13 11:37 AM
Lesson 6.4 Skills Practice
Name
Date
Making Some Assumptions
Inverse, Contrapositive, Direct Proof, and Indirect Proof
Vocabulary
Define each term in your own words.
1. inverse
2. contrapositive
3. direct proof
4. indirect proof (or proof by contradiction)
© Carnegie Learning
5. Hinge Theorem
6
6. Hinge Converse Theorem
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 617
617
21/06/13 11:39 AM
Lesson 6.4 Skills Practice
page 2
Problem Set
Write the converse of each conditional statement. Then, determine whether the converse is true.
1. If two lines do not intersect and are not parallel, then they are skew lines.
The converse of the conditional would be:
If two lines are skew lines, then they do not intersect and are not parallel.
The converse is true.
2. If two lines are coplanar and do not intersect, then they are parallel lines.
3. If a triangle has one angle whose measure is greater than 90º, then the triangle is obtuse.
6
5. If the lengths of the sides of a triangle measure 5 mm, 12 mm, and 13 mm, then it is a right triangle.
618 © Carnegie Learning
4. If a triangle has two sides with equal lengths, then it is an isosceles triangle.
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 618
17/06/13 11:37 AM
Lesson 6.4 Skills Practice
page 3
Name
Date
6. If the lengths of the sides of a triangle are 3 cm, 4 cm, and 5 cm, then the triangle is a right triangle.
7. If the corresponding sides of two triangles are congruent, then the triangles are congruent.
8. If the corresponding angles of two triangles are congruent, then the triangles are similar.
Write the inverse of each conditional statement. Then, determine whether the inverse is true.
9. If a triangle is an equilateral triangle, then it is an isosceles triangle.
The inverse of the conditional would be:
If a triangle is not an equilateral triangle, then it is not an isosceles triangle.
© Carnegie Learning
The inverse is not true.
10. If a triangle is a right triangle, then the sum of the measures of its acute angles is 90º.
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 619
6
619
17/06/13 11:37 AM
Lesson 6.4 Skills Practice
page 4
11. If the sum of the internal angles of a polygon is 180º, then the polygon is a triangle.
12. If a polygon is a triangle, then the sum of its exterior angles is 360º.
13. If two angles are the acute angles of a right triangle, then they are complementary.
14. If two angles are complementary, then the sum of their measures is 90º.
6
16. If a polygon is a trapezoid, then it is a quadrilateral.
620 © Carnegie Learning
15. If a polygon is a square, then it is a rhombus.
Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 620
17/06/13 11:37 AM
Lesson 6.4 Skills Practice
Name
page 5
Date
Write the contrapositive of each conditional statement. Then, determine whether the contrapositive is true.
17. If one of the acute angles of a right triangle measures 45º, then it is an isosceles right triangle.
The contrapositive of the conditional would be:
If a triangle is not an isosceles right triangle, then it is not a right triangle with an acute angle that
measures 45º.
The contrapositive is true.
18. If one of the acute angles of a right triangle measures 30º, then it is a 30º260º290º triangle.
19. If a quadrilateral is a rectangle, then it is a parallelogram.
© Carnegie Learning
20. If a quadrilateral is an isosceles trapezoid, then it has two pairs of congruent base angles.
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 621
621
17/06/13 11:37 AM
Lesson 6.4 Skills Practice
page 6
21. If the sum of the measures of two angles is 180º, then the angles are supplementary.
22. If two angles are supplementary, then the sum of their measures is 180º.
23. If the radius of a circle is 8 meters, then the diameter of the circle is 16 meters.
© Carnegie Learning
24. If the diameter of a circle is 12 inches, then the radius of the circle is 6 inches.
6
622 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 622
17/06/13 11:37 AM
Lesson 6.4 Skills Practice
page 7
Name
Date
Create an indirect proof to prove each statement.
____
____
____
25. Given: WY 
​ ​ bisects XYZ and XW 
​ ​  ZW 
​ ​
 
Y
___
___
Prove: XY 
​ ​  ZY 
​ ​ 
X
___ ___ Statements
1. ​XY ​  ZY 
____ ​ ​ 
2. ​WY ​ bisects XYZ
3.  XYW   ZYW
____ ____
4. ​YW ​  YW ​
​  
5.  XYW   ZYW
____ ____
6. ​XW ​  ZW ​
​  
____ ____
7. ​XW ​  ZW ​
​  
___ ___
8. ​XY ​  ZY 
​ ​ is false.
___ ___
​ ​ is true.
9. ​XY ​  ZY 
Z
Reasons
1. Assumption
2. Given
3. Definition of angle bisector
4. Reflexive Property of 
5. SAS Congruence Theorem
6. CPCTC
7. Given
8. Step 7 contradicts Step 6.
The assumption is false.
9. Proof by contradiction
26. Given: mEBX  mEBZ ___
Prove: EB 
​ ​ is
W
X
not an altitude of EZX.
B
E
Z
© Carnegie Learning
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 623
623
17/06/13 11:38 AM
Lesson 6.4 Skills Practice
page 8
___
27. Given: OMP  MOP and NP 
​ ​ does not bisect ONM.
M
____
____
Prove: NM 
​ ​  NO 
​ ​ 
P
N
O
___
___
___
___
​ ​ and EU 
​ ​  DU 
​ ​  
28. Given: ET 
​ ​  DT 
T
___
___
Prove: EX 
​ ​  DX 
​ ​ 
E
X
D
© Carnegie Learning
U
6
624 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 624
17/06/13 11:38 AM
Lesson 6.4 Skills Practice
page 9
Name
Date
For each pair of triangles, use the Hinge Theorem or its converse to write a conclusion using an inequality,
29.
P
30.
G
D
R
A
108°
N
S
68°
B
X
120°
F
Q
Q
SP . GQ
31.
P
3
32.
X
R
I
F
5
T
A
E
Z
C
4.5
3.5
U
© Carnegie Learning
K
6
Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 625
625
17/06/13 11:38 AM
© Carnegie Learning
6
626 Chapter 6 Skills Practice
451448_Skills_Ch06_587-626.indd 626
17/06/13 11:38 AM
1/--pages
Пожаловаться на содержимое документа