5.1 Congruence and Triangles - Murrieta Valley Unified School District
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5.1 Congruence and Triangles
Goal
Identify congruent triangles and corresponding parts.
Key Words
? corresponding parts ? congruent figures
The houses below are located in San Francisco, California. The two triangles outlined on the houses have the same size and shape.
E B
D
F
A
C
Suppose T ABC slides over to fit on T DEF. When the vertices are matched this way, a A and aD are corresponding angles and A&C* and D&F** are corresponding sides.
Corresponding angles and corresponding sides are examples of corresponding parts .
Figures are congruent if all pairs of corresponding angles are congruent and all pairs of corresponding sides are congruent.
Congruent
Not congruent
Student Help
LOOK BACK To review vertices of a triangle, see p. 175.
Same size and shape
Different sizes or shapes
EXAMPLE 1 List Corresponding Parts
Given that TJKL c TRST, list all corresponding congruent parts.
LT
Solution
J
KS
R
The order of the letters in the names of the triangles shows which parts correspond.
Corresponding Angles
Corresponding Sides
TJKL c TRST, so aJ c aR.
TJKL c TRST, so J&K* c R&S*.
TJKL c TRST, so aK c aS.
TJKL c TRST, so K&L* c S&T*.
TJKL c TRST, so aL c aT.
TJKL c TRST, so J&L* c R&T*.
5.1 Congruence and Triangles 233
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Student Help
STUDY TIP To name the triangles in Example 2, you can start at any vertex. TACB c TFED
TBCA c TDEF
TCAB c TEFD
Be sure the letters are listed in corresponding order.
EXAMPLE 2 Write a Congruence Statement
The two triangles are congruent. a. Identify all corresponding congruent parts. b. Write a congruence statement.
A
F
B
CD
E
Solution
a. Corresponding Angles aA c aF aB c aD aC c aE
Corresponding Sides A&B* c F&D* B&C* c D&E* A&C* c F&E*
b. List the letters in the triangle names so that the corresponding
angles match. One possible congruence statement is TABC c TFDE.
EXAMPLE 3 Use Properties of Congruent Triangles
In the diagram, TPQR c TXYZ. P a. Find the length of X&Z*.
P
Y
X
95
b. Find maQ.
10
RZ
Solution a. Because X&Z* c P&R*, you know that XZ PR 10. b. Because aQ c aY, you know that maQ maY 95.
Name Corresponding Parts and Congruent Triangles
1. Given TSTU c TYXZ, list all
2. Given T ABC c TDEF, find
corresponding congruent parts.
the length of D&F* and maB.
U X
S
TY
C 3
A
B
D
E
28
Z
F
3. Which congruence statement is correct? Why?
A. T JKL c TMNP
K
B. T JKL c TNMP
C. T JKL c TNPM
J
LN
M P
234 Chapter 5 Congruent Triangles
Page 3 of 7
Student Help
STUDY TIP Since E&F is a shared side of both triangles, you know that E&F c E&F by the Reflexive Property of Congruence.
EXAMPLE 4 Determine Whether Triangles are Congruent
Use the two triangles at the right.
a. Identify all corresponding
congruent parts.
E
b. Determine whether the triangles
are congruent. If they are congruent, write a congruence statement.
D
G
Solution
a. Corresponding Angles Corresponding Sides
aD c aG
D&E** c G&E*
aDEF c aGEF
D&F** c G&F*
aDFE c aGFE
E&F* c E&F*
F
b. All three sets of corresponding angles are congruent and all three
sets of corresponding sides are congruent, so the two triangles are congruent. A congruence statement is TDEF c TGEF.
Visualize It!
Remember that when parallel lines are cut by a transversal, the alternate interior angles are congruent.
L
K
J
H
G
EXAMPLE 5 Determine Whether Triangles are Congruent
In the figure, H&G** L&K*. Determine
L
whether the triangles are congruent.
If so, write a congruence statement.
K J
H
G
Solution
Start by labeling any information you can conclude from the figure. You can list the following angles congruent.
aHJG c aKJL aH c aK aG c aL
Vertical angles are congruent. Alternate Interior Angles Theorem Alternate Interior Angles Theorem
The congruent sides are marked on the diagram, so H&J* c K&J*, H&G** c K&L*, and J&G* c &JL*. Since all corresponding parts are congruent, THJG c TKJL.
Determine Whether Triangles are Congruent
4. In the figure, X&Y* Z&W**. Determine
X
whether the two triangles are
congruent. If they are, write a
congruence statement.
Y
W V
Z
5.1 Congruence and Triangles 235
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5.1 Exercises
Guided Practice
Vocabulary Check
Determine whether the angles or sides are corresponding angles, corresponding sides, or neither.
1. aC and aL
2. A&C* and J&K*
B
K
3. B&C* and K&L*
4. aB and aL
A
CJ
L
Skill Check
Given that TXYZ c TEFD, determine the congruent side or angle
that corresponds to the side or angle.
5. aY
6. aD
7. X&Z*
8. F&D*
In Exercises 9?12, find the missing length or angle measure,
given that TLMN c TPQR.
N
9. maP __?__
10. maM __?__
P 45 P
11.6
8.5
11. QR __?__
12. LN __?__
105
L6M
R
13. Determine whether the triangles are congruent. If they are, write a congruence statement.
E
K
F
GJ
L
Practice and Applications
Extra Practice
See p. 683.
Homework Help
Example 1: Exs. 14?19, 21?24
Example 2: Exs. 21?36 Example 3: Exs. 37?40 Example 4: Exs. 42?47 Example 5: Exs. 42?47
Corresponding Parts Determine whether the angles or sides are corresponding angles, corresponding sides, or neither.
14. aM and aS
15. aL and aR
M
S
16. M&N*** and S&T*
17. L&M** and T&S*
18. R&T* and L&N**
19. aN and aR
L
NR
T
20. Congruent Triangles Determine which of the following is a correct congruence statement for the triangles in Exercises 14?19. There is only one correct answer.
A. TLNM c TRTS B. TNML c TRST
C. TMNL c TSTR
236 Chapter 5 Congruent Triangles
Page 5 of 7
Identifying Congruent Parts In the diagram, TLMN c TPQR. Complete the statement with the corresponding congruent part.
21. aR c __?__
22. L&N* c __?__
L
P
23. aRPQ c __?__
24. Q&P* c __?__
25. TMNL c __?__ 26. TPRQ c __?__
M
P
NR
Naming Congruent Triangles Determine whether the congruence statement correctly describes the congruent triangles shown in the necklace.
27. TFGH c TNML
28. TGFH c TNLM
29. TGHF c TMLN 30. THFG c TLNM 31. THGF c TLNM
H
GM
L
32. TFHG c TLNM
F N
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.
P
S
34. A
B
F
C
P
RT
U
D
E
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.
CF
36.
B
A
C
E
A
BE
D
F
D
5.1 Congruence and Triangles 237
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