Reteach Triangle Congruence

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Name
LESSON
4-6
Date
Class
Reteach
Triangle Congruence: CPCTC
Corresponding Parts of Congruent Triangles are Congruent (CPCTC) is useful in proofs. If
you prove that two triangles are congruent, then you can use CPCTC as a justification for
proving corresponding parts congruent.
_ _
_
_
!
Given: AD CD , AB CB
$
Prove: A C
Proof:
#
!$#$
"
'IVEN
N!"$N#"$
!"#"
'IVEN
333
!#
#0#4#
"$"$
2EFLEX0ROPOF
Complete each proof.
_
,
_
1. Given: PNQ LNM, PN_
LN ,
N is the midpoint of QM .
_
0
-
_
Prove: PQ LM
Proof:
.
1
0.1,.Given
c. O01.
0.,.
a. Given
.is the
mdpt. of -1.
b. 1.
Given
Def. of midpt.
O,-.
d.CPCTC
SAS
-.
2. Given: UXW
and UVW are right s.
_
_
UX UV
5
Prove: X V
8
7
Proof:
Statements
Reasons
1. UXW and UVW are rt. s.
1. Given
_
_
2. UX UV
_
4. c.
2. a.
Given
3. b.
Reflex. Prop. of 4. d.
HL
5. e.
CPCTC
_
3. UW UW
UXW UVW
5. X V
Copyright © by Holt, Rinehart and Winston.
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01,-
46
6
Holt Geometry
Name
Date
Class
Reteach
LESSON
4-6
Triangle Congruence: CPCTC
continued
You can also use CPCTC when triangles are on the coordinate plane.
Given: C(2, 2), D(4, –2), E(0, –2),
F(0, 1), G(–4, –1), H(–4, 3)
Y
(
2
Prove: CED FHG
#
&
X
0
2
Step 1 Plot the points on a coordinate plane.
2
'
$
%
Step 2 Find the lengths of the sides of each triangle.
Use the Distance Formula if necessary.
d
2
2
x1) (y2 y1)
(x2
CD (4 2)
2
(2 2)2
FG EC GH 4
[2 (2)]
_ _
_
HF _
2
2
[0 (4)]
(1 3)
4 16 2 5
16 4 2 5
2
2
(2 0)
(1 1)
4 16 2 5
DE 4
2
2
(4 0)
_
= 16 4 2 5
_
So, CD FG , DE GH , and EC HF . Therefore CDE FGH by SSS, and
CED FHG by CPCTC.
Use the graph to prove each congruence statement.
Y
2
3
Y
"
9
3
2
#
! X
X
1
2
0
2
0
2
*
2
2 7
3
,
+
8
3. RSQ XYW
4. CAB LJK
QR WX 13 ,
AB JK 5, BC KL RS XY 7, SQ YW 34 .
10 , CA LJ 53 . So
So QRS WXY by SSS, and
ABC JKL by SSS, and
RSQ XYW by CPCTC.
CAB LJK by CPCTC.
5. Use the given set of points to prove PMN VTU.
M(–2, 4), N(1, –2), P(–3, –4), T(–4, 1), U(2, 4), V(4, 0)
MN TU 3 5 , NP UV 2 5 , PM VT 65 .
So MNP TUV by SSS, and PMN VTU by CPCTC.
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47
Holt Geometry
Name
LESSON
4-6
Date
Class
Name
Practice A
LESSON
4-6
Triangle Congruence: CPCTC
Parts
1. CPCTC is an abbreviation of the phrase “Corresponding
Triangles
of Congruent
Use the figure for Exercises
_2 and
_3.
�B � �E, �C � �F, and AB � DE .
�
�
�
�
�
3. Use CPCTC to name the other three pairs of congruent parts in the triangles.
_
�D
�
_
_
AC
DF
�
4. Some hikers come to a river in the woods. They
want to cross the river but decide to find out how
wide it is first. So they set up congruent right
triangles. The figure shows the river and the
triangles. Find the width of the river, GH.
BC
�
5m
�
3m
�
�
Prove: �P � �R
_
�
�
QS � QS
2. Reflexive Property of �
3. SAS
4. c.
CPCTC
Write a two-column proof.
�
XY �
3
BC �
YZ �
�
5
_
4
SSS
CPCTC
9. How do you know that �X � �A?
Name
LESSON
4-6
43
Date
Class
Holt Geometry
4. Reflex. Prop. of �
5. �FIH � �GHI
5. SAS
6. FH � GI
6. CPCTC
7. FH � GI
7. Def. of � segs.
LESSON
4-6
�
Prove: The diagonals of a parallelogram bisect each other.
_
�
2. Given: FGHI is a rhombus.
Prove: The diagonals of a rhombus are congruent, perpendicular,
and bisect the vertex angles of the rhombus.
_ _
_
�
�
Prove: �A � �C
Proof:
�
�������
�
�����
�����������
�������
�����
�
���
�������
�����
�������
�
�����������������
�
�
Complete each proof.
_ definition
Possible
_ _ answer: From the
_IH is congruent to
_ of a rhombus,
FG, IF is congruent to GH, and IH is parallel to FG. By Alternate Interior
Angles Theorem, �GFH is congruent to �IHF and �FGI is congruent_
by ASA. By CPCTC, FJ
to �HIG. Therefore
_ is congruent to �HIJ
_ �FGJ
_
is congruent to HJ and GJ is congruent to IJ. So �FJI is congruent to
�GHJ by SSS. But �HIJ is also congruent to �FIJ by SSS. And so all
four triangles are congruent by the Transitive_
Property of Congruence.
By
_
CPCTC and the Segment Addition Postulate, FH is congruent to GI. By
�FJI, �GJF, �HJG, and �IJH
CPCTC and the Linear_
Pair Theorem,
_
By CPCTC, �GFH, �
are right angles. So FH and GI are perpendicular.
_
so FH bisects �IFG and �IHG.
IFH, �GHF, and �IHF are congruent,
_
Similar reasoning shows that GI bisects �FGH and �FIH.
_
�
_
1. Given: �PNQ � �LNM, PN_
� LN ,
N is the midpoint of QM.
_
�
�
_
Prove: PQ � LM
Proof:
�
�
�����������
Given
�������
a. Given
��is the
mdpt. of ��.
Given
3. Rectangles, rhombuses, and squares are all types of parallelograms.
Write a conjecture about the diagonals of a rectangle.
b. ��
c. ����
� ��
�
Prove: �X � �V
�
The diagonals of a square are congruent perpendicular bisectors that
Statements
Reasons
1. �UXW and �UVW are rt. �s.
1. Given
5. An isosceles trapezoid has one pair of noncongruent
parallel sides, a pair of congruent nonparallel sides,
and two pairs of congruent angles. What relationship
do the diagonals of an isosceles trapezoid have?
_
2. UX � UV
2. a.
Given
3. UW � UW
3. b.
Reflex. Prop. of �
_
4. c.
_
�UXW � �UVW
5. �X � �V
The diagonals are congruent.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
77
�
�
bisect the vertex angles of the square.
_
�������
d.CPCTC
Def. of midpt.
Proof:
4. A square is a type of rhombus. Write a conjecture about the diagonals of a square.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
� ����
SAS
2. Given: �UXW
and �UVW are right �s.
_
_
UX � UV
The diagonals of a rectangle bisect each other.
45
Holt Geometry
Triangle Congruence: CPCTC
_
(Note: Be careful naming the triangles. The order of vertices matters.)
_
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Class
Reteach
Given: AD � CD, AB � CB
�
Possible
of a parallelogram, DC is congruent
_ From the definition
_ answer:
_
to AB and DC is parallel to AB . By the Alternate Interior Angles
Theorem, �BAC is congruent to �DCA and �CDB is congruent to _
�ABD. Therefore
is congruent to �CDE
_ by ASA. By CPCTC, DE
_�ABE
_
is congruent to BE and AE is congruent to CE. Congruent segments have
equal lengths, so the diagonals bisect each other.
�
Date
Corresponding Parts of Congruent Triangles are Congruent (CPCTC) is useful in proofs. If
you prove that two triangles are congruent, then you can use CPCTC as a justification for
proving corresponding parts congruent.
�
1. Given: ABCD is a parallelogram.
44
Name
Triangle Congruence: CPCTC
�
_
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Practice C
Write paragraph proofs for Exercises 1 and 2.
3. Rt. � � Thm.
_
4. IH � IH
_
4
XZ �
8. Name the triangle congruence theorem that shows �ABC � �XYZ.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
2. FI � GH, �FIH and �GHI are right angles. 2. Def. of rectangle
3. �FIH � �GHI
AC �
5
Reasons
1. Given
_
_
�
�
�
�
Statements
1.FGHI is a rectangle.
�
�
��
7. Use the Distance Formula to find the length of each side.
3
�
��
�
�
Prove: The diagonals of a rectangle have equal lengths. Possible answer:
�
6. Plot these points: A (0, 0), B (0, 3), C(4, 0),
X (�4, �3), Y(�4, 0), Z (0, �3). Draw
segments to make �ABC and �XYZ.
�
3. Given: FGHI is a rectangle.
�
Use the blank graph for Exercises 6–9.
�����������
���
�����������
�����
�������
������������������
�
AB �
�����������
�����������������
�������
�����
�PQS � �RQS
4. �P � �R
��� �����
������
Reasons
3. b.
�
�
�
1. Given
2. a.
�
_
2. Given: �L � �J, KJ � LM
Prove: �LKM � �JMK
_
�
Statements
_
� RQ,_
�PQS � �RQS
1. PQ
_
� 15 ft
25 ft
�
�
_
QS � QS,
CPCTC,
�PQS � �RQS
5. Given: PQ � RQ, �PQS � �RQS
15 ft
corresponds with CE, so Heike could not have jumped this distance.
Write a flowchart proof.
5m
_
�
20 ft
have jumped this distance. The distance along path CA is 25 ft because CA
�
�
_
�
distance along path BA is 20 ft because BA corresponds with DE, so Heike could
Use the phrases in the word bank to complete this proof.
_
Triangle Congruence: CPCTC
same length as its corresponding side in �EDC. Heike could jump about 23 ft. The
_
EF
�
3m
Practice B
Vertical � Thm. the triangles are congruent by ASA, and each side in �ABC has the
AAS
2. Name the triangle congruence theorem that shows �ABC � �DEF.
�A
Class
1. Heike Dreschler set the Woman’s World Junior Record for the
long jump in 1983. She jumped about 23.4 feet. The diagram
shows two triangles and a pond. Explain whether Heike
could have jumped the pond along path BA or along
path CA. Possible answer: Because �DCE � �BCA by the
are Congruent.”
�
Date
4. d.
HL
5. e.
CPCTC
46
Holt Geometry
Holt Geometry
Name
Date
Class
Name
Reteach
LESSON
4-6
LESSON
4-6
Triangle Congruence: CPCTC continued
You can also use CPCTC when triangles are on the coordinate plane.
Given: C(2, 2), D(4, –2), E(0, –2),
F (0, 1), G(–4, –1), H(–4, 3)
2
Prove: �CED � �FHG
�
�
�
0
�2
Step 1 Plot the points on a coordinate plane.
2
�
�
�
Step 2 Find the lengths of the sides of each triangle.
Use the Distance Formula if necessary.
CD �
�(4 � 2)
2
� (�2 � 2)
�
2
FG �
� (�1 � 1)
�
� �16 � 4 � 2 �5
GH � 4
DE � 4
EC �
��
2
2
�(�4 � 0)
�
�
� �4 � 16 � 2 �5
�
2
2
�(2 � 0)
_
_
� (1 � 3)
�
= �16 � 4 � 2 �5
�������
����������
_
So, CD � FG, DE � GH, and EC � HF . Therefore �CDE � �FGH by SSS, and
�CED � �FHG by CPCTC.
�
�
3
�
0
�2
2
�
�
� �
0
�2
�����
2
2
�
�2 �
�
3. �RSQ � �XYW
AB � JK � 5, BC � KL �
�
�
So �QRS � �WXY by SSS, and
�ABC � �JKL by SSS, and
�RSQ � �XYW by CPCTC.
�CAB � �LJK by CPCTC.
M(–2, 4), N(1, –2), P(–3, –4), T(–4, 1), U(2, 4), V(4, 0)
������������
���
�����
_
_
�
�
�
a. �QNT is isosceles.
So �MNP � �TUV by SSS, and �PMN � �VTU by CPCTC.
47
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All rights reserved.
Name
Date
Holt Geometry
Problem Solving
LESSON
4-6
1. Two triangular plates are congruent. The area of one of the plates is 60 square inches.
What is the area of the other plate? Explain.
2
triangles also have the same areas.
_
_
38 m
�
3. A city planner sets up the triangles to find
the distance RS across a river. Describe
the steps that she can use to find RS.
�
76 m
�
38 m
�
���������
�������������
�
�����
_
���������
�����
rt. �. PQ � RQ because
1. What are some reasons you would use an acronym?
�
PQ � RQ � 65 ft. �NQP �
Possible answer: to abbreviate or make a statement simpler to
understand or remember
�SQR because vert. � are �. Therefore �NPQ � �SRQ by ASA.
_
2. What are some other acronyms you have used in your everyday life?
By CPCTC, NP � SR. So SR � NP � 40 ft.
Choose the best answer.
Answers will vary. Students may mention FBI, IRS, RSVP, FAQ, or
�
others that are popular in text messaging.
4. A lighthouse and the range of its shining light are
shown. What can you conclude?
A x � y by CPCTC
C �AED � �ADE by CPCTC
B x � 2y
D �AED � �ACB
Examine the figure and answer the question.
�
5. A rectangular piece of cloth 15 centimeters
long is cut along a diagonal to form two
triangles. One of the triangles has a side
length of 9 centimeters. Which is a true
statement?
����
�
�
����
6. Small sandwiches are cut in the
shape of right triangles. The longest
sides of all the sandwiches are 3 inches.
One sandwich has a side length of
2 inches. Which is a true statement?
A All the sandwiches have a side
length of 2 inches by CPCTC.
G The second triangle has a side
length of 9 centimeters by CPCTC.
B All the sandwiches are isosceles
triangles with side lengths of 2 inches.
H You cannot make a conclusion about
the side length of the second triangle.
C None of the other sandwiches have
side lengths of 2 inches.
J The triangles are not congruent.
D You cannot make a conclusion using
CPCTC.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
�
F The second triangle has an angle
measure of 15° by CPCTC.
49
���������
�������������
��������
�
�����
Holt Geometry
Using an Acronym
�����
�
Class
Reading Strategies
�
�P � �R because they are both
Date
One acronym used in geometry is CPCTC.
Look at the breakdown of this acronym:
82 m
76 m
by CPCTC. Therefore UV � XY � 82 m.
c. �VMN is isosceles.
An acronym is a word formed from the first letters of a phrase. For
example, ASAP stands for “As Soon As Possible.” Acronyms can also
combine the first letters or series of letters in a series of words, as in
radar, which stands for radio detecting and ranging.
60 in ; Since the triangles are �, they have the same measures. So, the
�
48
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All rights reserved.
Name
Triangle Congruence: CPCTC
82 m; �UVW � �XYW by SAS, so UV � XY
b. �MPV is isosceles.
5. Given that m�NQT � 110°, find the measures of all the other angles between the rafters,
beams, and struts of the queen-post truss. Label the angle measures directly on the figure.
Class
2. An archaeologist draws the triangles to find the distance
XY across a ravine. What is XY ? Explain.
�
4. Using the information about the queen-post truss given above, prove each statement
on a separate sheet of paper. Use any form of proof that you want. Proofs will vary.
MN � TU � 3 � 5 , NP � UV � 2 �5 , PM � VT � �65 .
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
������������
�������� ����������������������
���������
������
��������
��������
����������������
������
��� � ���
In the_
queen-post truss pictured at right, congruent queen posts NV
��� � ���� � ���
so that_
they are perpendicular
to_
tie beam
and
_ TW are positioned
_
_
_ _
��� ���
PR and so that PV_
� VW_
� WR.
Struts _
MV, SW,
NT intersect �
�
_
_ and
_
��� ��� ��� ���
The
the rafters so that PM � MN � NQ and RS � ST � TQ.
����
��� ��� ����
_
�
�
�
outer triangle, �PQR, is an isosceles triangle with base PR.
��� ���
��� ���
5. Use the given set of points to prove �PMN � �VTU.
_
�
3. Given that m�KJZ � 33°, find the measures of all the other angles between the rafters,
beams, and struts of the king-post truss. Label the angle measures directly on the figure.
�
�10, CA � LJ � �53 . So
_
�
������
�
���
���
���� ��� ��� ����
��� ���
��� ���
�
4. �CAB � �LJK
�
RS � XY � 7, SQ � YW � � 34 .
4-6
�
��������
Explanations may vary.
�
QR � WX � �13 ,
LESSON
�
2. Explain why it must be true that �XJZ � �XZJ. (Hint: How is ZX related to �JKL?)
�3
�
������
����������������
��������
��������
�������
��������
�
�
�
�
������
�����������
�����
Use the graph to prove each congruence statement.
�
Isosceles Triangles and Roof Trusses
1. Refer to the diagram of the king-post truss. Write a flowchart proof to show that
�JKZ � �LKZ.
�
2
2
�[0 � (�4)]
�
�
_ _
_
HF �
� [2 � (�2)]
�
� �4 � 16 � 2 �5
Challenge
For example, a_
king-post truss is pictured at right.
The king post, KZ, is a median of �JKL, and it provides
support for the rafters.
support for the rafters
_ Additional
_
comes from struts ZX and ZY , which are medians of
�JKZ and �LKZ, respectively. The outer_
triangle,
�
�JKL, is an isosceles triangle with base JL.
�
2
2
� x1) � (y2 � y1)
�(x2�
d�
Class
The wooden or metal framework that supports a roof is called a
roof truss. The simplest type of roof truss has the shape of an
isosceles triangle, as depicted
by_
�ABC at right. In the diagram,
_
the legs of the triangle, AB and
_CB, represent sloping beams that
are called rafters. The base, AC, represents the tie beam that
“ties together” the rafters. However, large roofs require trusses
with designs that are more complex than this.
�
�
Date
�
�
�
�
�
_
_
3. In this triangle, �C � �N and AC � LN. Assume that �ABC � �LMN.
Name four other parts that are congruent using CPCTC.
_
_ _
_
�B � �M ; �A � �L; CB � NM; AB � LM
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50
Holt Geometry
Holt Geometry
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