THINK AND DISCUSS
−− −− −−− −−
1. In the figure, UV XY, VW YZ,
and ∠V ∠Y. Explain why
UVW XYZ. By CPCTC, which
additional parts are congruent?
1
8
6
7
9
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2. GET ORGANIZED Copy and complete the graphic organizer.
Write all conclusions you can make using CPCTC.
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4-6
Exercises
KEYWORD: MG7 4-6
KEYWORD: MG7 Parent
GUIDED PRACTICE
1. Vocabulary You use CPCTC after
proving triangles are congruent.
Which parts of congruent triangles
are referred to as corresponding parts?
SEE EXAMPLE
1
p. 260
2. Archaeology An archaeologist
wants to find the height AB of a
rock formation. She places a
marker at C and steps off the
distance from C to B. Then she
walks the same distance from
C and places a marker at D.
If DE = 6.3 m, what is AB?
SEE EXAMPLE
p. 260
2
−− −− −−
3. Given: X is the midpoint of ST. RX ⊥ ST
−− −−
Prove: RS RT
Proof:
262
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Chapter 4 Triangle Congruence
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SEE EXAMPLE
3
p. 261
−− −− −− −−
4. Given: AC AD, CB DB
−−
Prove: AB bisects ∠CAD.
Proof:
Statements
−− −−− −− −−
1. AC AD, CB DB
?
−−−−
3. ACB ADB
2. b.
4. ∠CAB ∠DAB
−−
5. AB bisects ∠CAD
SEE EXAMPLE 4
p. 261
Reasons
1. a. ?
−−−−
2. Reflex. Prop. of ?
−−−−
?
−−−−
5. e. ?
−−−−
3. c.
4. d.
Multi-Step Use the given set of points to prove each congruence statement.
5. E(-3, 3), F(-1, 3), G(-2, 0), J(0, -1), K(2, -1), L(1, 2); ∠EFG ∠JKL
6. A(2, 3), B(4, 1), C(1, -1), R(-1, 0), S(-3, -2), T(0, -4); ∠ACB ∠RTS
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
7
8–9
10–11
12–13
1
2
3
4
Extra Practice
Skills Practice p. S11
Application Practice p. S31
7. Surveying To find the distance AB across
a river, a surveyor first locates point C.
He measures the distance from C to B.
Then he locates point D the same distance
east of C. If DE = 420 ft, what is AB?
A
B
500 ft
500 ft C
D
E
8. Given: M is the midpoint of
−−
−−
PQ and RS.
−− −−
Prove: QR PS
,
*
−−− −− −− −−−
9. Given: WX XY YZ ZW
Prove: ∠W ∠Y
7
8
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9
+
-
−−
10. Given: G is the midpoint of FH.
−− −−
EF EH
−−−
−− −−
11. Given: LM bisects ∠JLK. JL KL
−−
Prove: M is the midpoint of JK.
Prove: ∠1 ∠2
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Multi-Step Use the given set of points to prove each congruence statement.
12. R(0, 0), S(2, 4), T(-1, 3), U(-1, 0), V(-3, -4), W(-4, -1); ∠RST ∠UVW
13. A(-1, 1), B(2, 3), C(2, -2), D(2, -3), E(-1, -5), F(-1, 0); ∠BAC ∠EDF
−−
14. Given: QRS is adjacent to QTS. QS bisects ∠RQT. ∠R ∠T
−−
−−
Prove: QS bisects RT.
−−
−−
15. Given: ABE and CDE with E the midpoint of AC and BD
−− −−
Prove: AB CD
4-6 Triangle Congruence: CPCTC
263
16. This problem will prepare you for the Multi-Step Test
Prep on page 280.
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The front of a doghouse has the dimensions shown.
a. How can you prove that ADB ADC?
−− −−
b. Prove that BD CD.
−−
−−
c. What is the length of BD and BC to the nearest tenth?
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Multi-Step Find the value of x.
17.
18.
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Use the diagram for Exercises 19–21.
19. Given: PS = RQ, m∠1 = m∠4
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Prove: m∠3 = m∠2
20. Given: m∠1 = m∠2, m∠3 = m∠4
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+
Prove: PS = RS
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21. Given: PS = RQ, PQ = RS
−− −−
Prove: PQ RS
22. Critical Thinking Does the diagram contain
enough information to allow you to conclude
−− −−−
that JK ML? Explain.
23. Write About It Draw a diagram and explain how a surveyor can set up triangles
to find the distance across a lake. Label each part of your diagram. List which
sides or angles must be congruent.
24. Which of these will NOT be used as a reason in a proof
−− −−
of AC AD?
SAS
ASA
CPCTC
Reflexive Property
25. Given the points K(1, 2), L(0, -4), M(-2, -3), and N(-1, 3),
which of these is true?
∠KNL ∠MNL
∠LNK ∠NLM
26. What is the value of y?
10
20
∠MLN ∠KLN
∠MNK ∠NKL
35
85
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27. Which of these are NOT used to prove angles congruent?
congruent triangles
parallel lines
noncorresponding parts
perpendicular lines
264
Chapter 4 Triangle Congruence
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