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ChapterChapter 55CongruentTrianglesCongruentTriangles
230
Where should a goalie
stand?
A soccer goalie instinctively imagines a triangle formed by the goal posts andthe ball. The best position to standallows the goalie to reach each side ofthe triangle in the same amount of time.
As the ball moves and the shape of the triangle changes, the goalie’s bestposition also changes.
Learn More About ItYou will learn more about soccer inExercises 21 and 22 on p. 278.
G
cgpe-0500 8/16/01 9:45 AM Page 230
Who usesCongruent Triangles?
How will you use these ideas?
• See how triangles are used in quilting. (p. 247)• Investigate origami patterns. (p. 255)• Identify triangles in skateboard ramps. (p. 257)• Analyze lighting for a stage production. (p. 262)• Look for patterns in string designs. (p. 269)• Find out how kaleidoscope patterns are created. (p. 285)
TYPE DESIGNERType designers design fonts that appear in books,magazines, newspapers, and other materials. Jonathan Hoefler hasdesigned many fonts that are widely used today. (p. 288)
FACILITIES PLANNERFacilities planners helpbusinesses determine the best locations for new buildings. They canhelp companies save money and run moreefficiently. (p. 275)
231
LAHGE_10_CSse_cp_230-231.qxd 4/16/09 4:30 PM Page 231
232 Chapter 5 Congruent Triangles
Chapter Study GuideStudy Guide55
What’s the chapter about?
• Identifying corresponding parts of congruent triangles• Showing triangles are congruent• Using angle bisectors and perpendicular bisectors• Using reflections and line symmetry
Key Words
Take this quick quiz. If you are unsure of an answer, look at the
reference pages for help.
Vocabulary Check (refer to p. 192)
1. Which segment in the figure is the hypotenuse?
�A KL&* �B LN&* �C NM&& �D KM&**
Skill Check (refer to pp. 61, 128)
2. If maFGH � 100° and GJ&*(bisects aFGH, what is maFGJ ?
�F 25° �G 50° �H 90° �J 100°
3. Use the diagram at the right to determine which ofthe following statements is true.
�A a7 c a2 �B a5 c a2
�C a1 c a8 �D a3 c a7
PREVIEW
L
K
M
N
7865
3421
PREVIEW
Chapter Readiness QuizPREPAREPREPARE
Studying TrianglesVISUAL STRATEGYVISUAL STRATEGY
Visualize It!
• corresponding parts, p. 233• congruent figures, p. 233• proof, p. 243• distance from a point to a line,
p. 273
• equidistant, p. 273• perpendicular bisector, p. 274• reflection, p. 282• line of symmetry, p. 284
When working with overlappingtriangles, you may find it helpful to draw the triangles separately.
cgpe-05sg 07/24/2001 7:48 PM Page 232
The houses below are located in San Francisco, California. The twotriangles outlined on the houses have the same size and shape.
Suppose TABC slides over to fit on TDEF. When the vertices are
matched this way, aA and aD are corresponding angles and AC&*
and DF&** are corresponding sides.
Corresponding angles and corresponding sides are examples of.
Figures are if all pairs of corresponding angles arecongruent and all pairs of corresponding sides are congruent.
Congruent Not congruent
congruent
corresponding parts
5.1 Congruence and Triangles 233
Goal
Identify congruenttriangles andcorresponding parts.
Key Words
• corresponding parts
• congruent figures
5.15.1 Congruence and Triangles
Given that TJKLcTRST, list allcorresponding congruent parts.
Solution
The order of the letters in the names of the triangles shows which parts correspond.
Corresponding Angles Corresponding Sides
TJKLcTRST, so aJc aR. TJKLcTRST, so JK&* c RS&*.
TJKLcTRST, so aKc aS. TJKLcTRST, so KL&* c ST&*.
TJKLcTRST, so aLc aT. TJKLcTRST, so JL&* c RT&*.
J K
L
S R
T
EXAMPLE 111 List Corresponding Parts
LOOK BACK
To review vertices of atriangle, see p. 175.
Student Help
Same size and shape Different sizes or shapes
B
A C
E
D FF
cgpe-0501 07/24/2001 7:46 PM Page 233
1. Given TSTUcTYXZ, list all 2. Given TABCcTDEF, findcorresponding congruent parts. the length of DF&* and maB.
3. Which congruence statement is correct? Why?
A.TJKLcTMNP
B. TJKLcTNMP
C.TJKLcTNPM LJ
K
PN
M
BA
ED
C
F
28�
3
TS
U
Y
X
Z
234 Chapter 5 Congruent Triangles
The two triangles are congruent.
a. Identify all corresponding congruent parts.
b. Write a congruence statement.
Solution
a. Corresponding Angles Corresponding Sides
aAc aF AB&* c FD&*
aBc aD BC&* c DE&*
aC c aE AC&* c FE&*
b. List the letters in the triangle names so that the corresponding angles match. One possible congruence statement is TABCcTFDE.
EXAMPLE 222 Write a Congruence Statement
In the diagram, TPQRcTXYZ.
a. Find the length of XZ&*.
b. Find maQ.
Solution
a. Because XZ&* c PR&*, you know that XZ � PR � 10.
b. Because aQc aY, you know that maQ � maY � 95�.
R Z
P Y XP 95�
10
EXAMPLE 333 Use Properties of Congruent Triangles
Name Corresponding Parts and Congruent Triangles
STUDY TIP
To name the trianglesin Example 2, you canstart at any vertex.
TACBcTFED
TBCAcTDEF
TCABcTEFD
Be sure the letters arelisted in correspondingorder.
Student Help B C
A
D E
F
cgpe-0501 07/24/2001 7:46 PM Page 234
5.1 Congruence and Triangles 235
Use the two triangles at the right.
a. Identify all corresponding congruent parts.
b. Determine whether the triangles are congruent. If they are congruent,write a congruence statement.
Solution
a. Corresponding Angles Corresponding Sides
aDc aG DE&** c GE&*
aDEFc aGEF DF&** c GF&*
aDFEc aGFE EF&* c EF&*
b. All three sets of corresponding angles are congruent and all three sets of corresponding sides are congruent, so the two triangles arecongruent. A congruence statement is TDEFcTGEF.
EXAMPLE 4 Determine Whether Triangles are Congruent
In the figure, HG&** � LK&*. Determine whether the triangles are congruent. If so, write a congruence statement.
Solution
Start by labeling any information you can conclude from the figure. You can list the following angles congruent.
aHJGc aKJL Vertical angles are congruent.
aHc aK Alternate Interior Angles Theorem
aG c aL Alternate Interior Angles Theorem
The congruent sides are marked on the diagram, so HJ&* c KJ&*, HG&** c KL&*,
and JG&*c JL&*. Since all corresponding parts are congruent, THJGcTKJL.
GH
J
L K
EXAMPLE 5 Determine Whether Triangles are Congruent
Determine Whether Triangles are Congruent
Remember that whenparallel lines are cut by a transversal, thealternate interior anglesare congruent.
J
GH
L K
Visualize It!
STUDY TIP
Since EF& is a sharedside of both triangles, you know that EF& c EF&by the Reflexive Propertyof Congruence.
Student Help
4. In the figure, XY&* � ZW&**. Determine whether the two triangles are congruent. If they are, write a congruence statement.
ZY
V
X W
D G
E
F
LAHGE_10_CSse_cp_233-239 4/23/09 2:31 AM Page 235
236 Chapter 5 Congruent Triangles
Determine whether the angles or sides are corresponding angles,
corresponding sides, or neither.
1. aC and aL 2. AC&* and JK&*
3. BC&* and KL&* 4. aB and aL
Given that TXYZ cTEFD, determine the congruent side or angle
that corresponds to the side or angle.
5. aY 6. aD 7. XZ&* 8. FD&*
In Exercises 9–12, find the missing length or angle measure,
given that TLMN c TPQR.
9. maP � __?__ 10. maM � __?__
11. QR � __?__ 12. LN � __?__
13. Determine whether the triangles are congruent. If they are, write a congruence statement.
Corresponding Parts Determine whether the angles or sides are
corresponding angles, corresponding sides, or neither.
14. aM and aS 15. aL and aR
16. MN&*** and ST&* 17. LM&** and TS&*
18. RT&* and LN&** 19. aN and aR
20. Congruent Triangles Determine which of the following is a correctcongruence statement for the triangles in Exercises 14–19. There isonly one correct answer.
A.TLNMcTRTS B. TNMLcTRST C. TMNLcTSTR
L
M S
N TR
Practice and Applications
L
K
JF
E
G
Skill Check
Vocabulary Check
Guided Practice
Exercises5.15.1
Extra Practice
See p. 683.
Example 1: Exs. 14–19,21–24
Example 2: Exs. 21–36Example 3: Exs. 37–40Example 4: Exs. 42–47Example 5: Exs. 42–47
Homework Help
M RL
N PP
105�
45�
8.5
6
11.6
LJ
K
CA
B
cgpe-0501 2/6/02 10:37 AM Page 236
Identifying Congruent Parts In the diagram, TLMN c TPQR.
Complete the statement with the corresponding congruent part.
21. aRc __?__
22. LN&* c __?__
23. aRPQc __?__
24. QP&* c __?__
25. TMNLc __?__
26. TPRQc __?__
Naming Congruent Triangles Determine whether the congruence
statement correctly describes the congruent triangles shown
in the wooden floor.
27. TFGHcTNML
28. TGFHcTNLM
29. TGHFcTMLN
30. THFGcTLNM
31. THGFcTLNM
32. TFHGcTLNM
Writing Congruent Statements In Exercises 33 and 34, the
triangles are congruent. Identify all pairs of corresponding
congruent parts.Then write a congruence statement.
33. 34.
Marking Triangles In Exercises 35 and 36, TABC c TDEF. Draw
the triangles on your paper.Then mark every angle and side of
the triangles to show the corresponding congruent parts.
35. 36.
F
B
E
D
A C
E
F
DA
C
B
A
D
C
E
F
BP
RP
S
UT
5.1 Congruence and Triangles 237
M
L
N
P
R
P
NN
MM
LL
GGFF
HH
LAHGE_10_CSse_cp_233-239 4/28/09 3:44 PM Page 237
Using Congruent Triangles In Exercises 37–40, TABC c TDEF.
37. Find the length of DF&* 38. Find the length of AB&*and maB. and maF.
39. Find the length of AB&* 40. Find the length of BC&*and maF. and maD.
41. Writing Congruence Statements Given that TFGH cTTSR,determine which congruence statement does not describe the triangles.
A.THGFcTRST B. TFHGcTTRS C. TGFHcTSRT
Congruent Triangles Determine whether the triangles are congruent.
If so, write a congruence statement.
42. 43.
44. 45.
46. 47. Z
W YXM
J K
L
DA
C
BER
T
SP
K
L M
J P N60�
30� 30�
60�
5 5
A
C
B
D
E
F
C
BD F
A E86�
27� 115
B
A F E
C D
30�
50�
4
6
B
A F E
C D39�
14
B
A D
C F E100�
5
238 Chapter 5 Congruent Triangles
HOMEWORK HELP
Extra help with problemsolving in Exs. 42–47 is at classzone.com
IStudent HelpI C L A S S Z O N E . C O M
cgpe-0501 07/24/2001 7:46 PM Page 238
Sculpture The diagram below represents a view of the base of a stainless
steel sculpture involving four triangles.
48. Explain how you know that aGJFc aHJF.
49. Explain how you know that aJGFc aJHF.
50. Do you have enough information to show that TGFJcTHFJ? Explain your reasoning.
Error Analysis In Exercises 51 and 52, refer to Jillian’s work and the
diagram below.
51. How does Jillian know that aBACc aDAE?
52. Is TABCcTADE? Explain your reasoning.
53. Multiple Choice Which congruence statementdoes not describe the triangles at the right?
�A TTSRcTXYZ �B TRSTcTXYZ
�C TSTRcTYXZ �D TTRScTXZY
Naming Properties Name the property that the statement
illustrates. (Lesson 2.6)
54. maJ � maJ 55. PQ&* c QP&*
56. If aPQRc aFGH, 57. If AB � CD and CD � FG, then aFGHc aPQR. then AB � FG.
Classifying Triangles Classify the triangle with the given side lengths
as acute, right, or obtuse. (Lesson 4.5)
58. 5, 2, 6 59. 5, 9, 10 60. 14, 5, 12
Evaluating Decimals Evaluate. (Skills Review, p. 655)
61. 4.15 � 7.8 62. 12.64 � 8.92 63. 14 � 5.33
64. 0.91 � 0.17 65. 1.476 � 6.2 � 9.4 66. 10.872 � 7.65
Algebra Skills
Mixed Review
Standardized Test
Practice
5.1 Congruence and Triangles 239
aABC c aADEaACB c aAEDaBAC c aDAE
The corresponding angles are congruent so TABC c TADE.
GJ
H
F
Z
XT Y
R
S
B
E
D
AC
Jillian
LAHGE_10_CSse_cp.qxd 4/5/09 2:00 PM Page 239