Congruent Triangles

Congruent Triangles
• Name and label corresponding parts of congruent triangles.
• Identify congruence transformations.
Vocabulary
• congruent triangles
• congruence
transformations
are triangles
used in bridges?
In 1930, construction started on
the West End Bridge in Pittsburgh,
Pennsylvania. The arch of the bridge
is trussed, not solid. Steel rods are
arranged in a triangular web that lends
structure and stability to the bridge.
CORRESPONDING PARTS OF CONGRUENT TRIANGLES Triangles
that are the same size and shape are congruent triangles . Each triangle has three
angles and three sides. If all six of the corresponding parts of two triangles are
congruent, then the triangles are congruent.
B
F
C
G
A
E
←
←
If nABC is congruent to nEFG, the vertices of the two triangles correspond in the
same order as the letters naming the triangles.
←
←
←
←
nABC > nEFG
This correspondence of vertices can be used to name the corresponding congruent
sides and angles of the two triangles.
/A > /E
/B > /F
/C > /G
AB
w
w>E
wF
w
B
C>F
ww
wG
w
A
C>E
ww
wG
w
The corresponding sides and angles can be determined from any congruence
statement by following the order of the letters.
Definition of Congruent Triangles (CPCTC)
Study Tip
Congruent Parts
In congruent triangles,
congruent sides are
opposite congruent
angles.
Two triangles are congruent if and only if their corresponding parts are congruent.
CPCTC stands for corresponding parts of congruent triangles are congruent. “If and
only if” is used to show that both the conditional and its converse are true.
192 Chapter 4 Congruent Triangles
Aaron Haupt
Example 1 Corresponding Congruent Parts
FURNITURE DESIGN The seat and legs of this
stool form two triangles. Suppose the measures
in inches are QR 5 12, RS 5 23, QS 5 24, RT 5 12,
TV 5 24, and RV 5 23.
a. Name the corresponding congruent angles and
sides.
/Q > /T
/QRS > /TRV
/S > /V
QR
w
w>T
wR
w
R
RV
wS
w>w
w
Q
T
R
S
Q
wS
w>T
wV
w
V
b. Name the congruent triangles.
nQRS > nTRV
Like congruence of segments and angles, congruence of triangles is reflexive,
symmetric, and transitive.
Properties of Triangle Congruence
Theorem 4.4
Congruence of triangles is reflexive, symmetric, and transitive.
Reflexive
nJKL ù nJKL
K
Transitive
If nJKL ù nPQR, and
nPQR ù nXYZ, then
nJKL ù nXYZ
K
L
J
Symmetric
If nJKL ù nPQR,
then nPQR ù nJKL.
K
L
J
Q
K
L
J
Q
R
Y
L
P
J
R
P
Z
X
You will prove the symmetric and reflexive parts of Theorem 4.4 in Exercises 33 and 35, respectively.
Proof
Theorem 4.4 (Transitive)
Given: nABC > nDEF
B
E
H
nDEF > nGHI
Prove:
nABC > nGHI
Proof:
Statements
1. nABC > nDEF
C
A
2. /A > /D, /B > /E, /C > /F
F
I
G
D
Reasons
1. Given
2. CPCTC
AB
DF
w
w>D
wE
w, B
wC
w>E
wF
w, A
wC
w>w
w
3. nDEF > nGHI
3. Given
4. /D > /G, /E > /H, /F > /I
4. CPCTC
DE
GIw
w
w>G
wH
w, E
wF
w>H
wIw, D
wF
w>w
5. /A > /G, /B > /H, /C > /I
5. Congruence of angles is transitive.
B>G
GIw
6. w
Aw
wH
w, B
wC
w>H
wIw, A
wC
w>w
6. Congruence of segments is transitive.
7. nABC > nGHI
7. Def. of > ns
www.geometryonline.com/extra_examples
Lesson 4-3 Congruent Triangles 193
Private Collection/Bridgeman Art Library
Study Tip
Naming
Congruent
Triangles
IDENTIFY CONGRUENCE TRANSFORMATIONS In the figures below,
nABC is congruent to nDEF. If you slide nDEF up and to the right, nDEF is still
congruent to nABC.
E'
B
There are six ways to
name each pair of
congruent triangles.
E
slide
A
D'
D
F'
C
F
The congruency does not change whether you turn nDEF or flip nDEF. nABC is
still congruent to nDEF.
E
D'
Study Tip
E
D
Transformations
Not all of the
transformations preserve
congruence. Only
transformations that do
not change the size or
shape of the triangle are
congruence
transformations.
flip
D'
turn
F
D
F'
F
F'
E'
E'
If you slide, flip, or turn a triangle, the size and shape do not change. These three
transformations are called congruence transformations .
Example 2 Transformations in the Coordinate Plane
COORDINATE GEOMETRY The vertices of nCDE
are C(25, 7), D(28, 6), and E(23, 3). The vertices
of nC9D9E9 are C9(5, 7), D9(8, 6), and E9(3, 3).
a. Verify that nCDE > nC9D9E9.
Use the Distance Formula to find the length of
each side in the triangles.
(25)]2w
2 7)2
1 (6w
[28 2w
DC 5 Ïw
5 Ï9w
1 1 or Ïw
10
1 (6w
DE 5 Ïw
[28 2w
(23)]2w
2 3)2
5 Ï25
1 9 or Ïw
34
w
1 (7w
CE 5 Ïw
[25 2w
(23)]2w
2 3)2
5 Ï4w
1 16 or Ïw
20
D
C
8
y
C'
D'
4
E
–8
–4
E'
O
4
8x
D9C9 5 Ï(8
2 5w
)2 1 (6w
2 7)2
w
5 Ï9w
1 1 or Ïw
10
D9E9 5 Ï(8
2 3w
)2 1 (6w
2 3)2
w
5 Ï25
1 9 or Ï34
w
w
C9E9 5 Ï(5
2 3w
)2 1 (7w
2 3)2
w
5 Ï4w
1 16 or Ï20
w
By the definition of congruence, w
DC
D9wC
DE
C9wE
w>w
w9w, w
w>D
w9wE
w9w, and C
wE
w>w
w9w.
Use a protractor to measure the angles of the triangles. You will find that the
measures are the same.
D9wE
CE
C9wE
In conclusion, because D
wC
w>D
w9wC
w9w, D
wE
w>w
w9w, and w
w>w
w9w, /D > /D9,
/C > /C9, and /E > /E9, nDCE > nD9C9E9.
b. Name the congruence transformation for nCDE and nC9D9E9.
nC9D9E9 is a flip of nCDE.
194 Chapter 4 Congruent Triangles
Concept Check
Guided Practice
1. Explain how slides, flips, and turns preserve congruence.
2. OPEN ENDED Draw a pair of congruent triangles and label the congruent sides
and angles.
Identify the congruent triangles in each figure.
D
3. A
4. H
K
F
C
B
J
T
5. If nWXZ > nSTJ, name the congruent angles and congruent sides.
B
6. QUILTING In the quilt design, assume that angles and
segments that appear to be congruent are congruent.
Indicate which triangles are congruent.
M
A
G
E
7. The coordinates of the vertices of nQRT and nQ9R9T9
are Q(24, 3), Q9(4, 3), R(24, 22), R9(4, 22), T(21, 22),
and T9(1, 22). Verify that nQRT > nQ9R9T9. Then name
the congruence transformation.
Application
8. GARDENING This garden lattice will be covered with
morning glories in the summer. Wesley wants to save
two triangular areas for artwork. If nGHJ > nKLP,
name the corresponding congruent angles and sides.
L
N
J
H
F
C
K
D
H
G
L
J
K
P
Practice and Apply
For
Exercises
See
Examples
9–22,
27–35
23–26
1
Identify the congruent triangles in each figure.
9.
10.
K
S
L
F
2
J
Extra Practice
See page 761.
V
H
C
R
P
11.
T
12.
Z
F
E
S
W
G
Q
V
H
Name the congruent angles and sides for each pair of congruent triangles.
13. nTUV > nXYZ
14. nCDG > nRSW
15. nBCF > nDGH
16. nADG > nHKL
Lesson 4-3 Congruent Triangles 195
Assume that segments and angles that appear to be congruent in the numbered
triangles are congruent. Indicate which triangles are congruent.
17.
18.
19.
5
9
2
1
5
4
3
6
7
1
9
8
6
10
7
8
1
2
10
2
3
11
4
12
12
7
3
8
6
11
4
10
14
13
17
18
16
15
19
20
9
5
20. All of the small triangles in the figure at the right
are congruent. Name three larger congruent
triangles.
S
V
B
U
F
E
T
A
C
D
21. MOSAICS The picture at the left is the center of a Roman mosaic. Because the
four triangles connect to a square, they have at least one side congruent to a side
in another triangle. What else do you need to know to conclude that the four
triangles are congruent?
Verify that each of the following preserves congruence and name the congruence
transformation.
22. nPQV > nP9Q9V9
23. nMNP > nM9N9P9
P
M
y Q'
Q
N
y
P'
P
V
Mosaics
A mosaic is composed of
glass, marble, or ceramic
pieces often arranged in
a pattern. The pieces, or
tesserae, are set in cement.
Mosaics are used to
decorate walls, floors,
and gardens.
O
V'
x
P'
M'
O
N'
x
24. nGHF > nG9H9F9
25. nJKL > nJ9K9L9
H
y
J
H'
y
L
L'
F
Source: www.dimosaico.com
G
F'
O
G'
x
K'
O
x
K
J'
Determine whether each statement is true or false. Draw an example or
counterexample for each.
26. Two triangles with corresponding congruent angles are congruent.
27. Two triangles with angles and sides congruent are congruent.
28. UMBRELLAS Umbrellas usually have eight congruent
triangular sections with ribs of equal length. Are the
statements nJAD > nIAE and nJAD > nEAI both
correct? Explain.
F
A
North Carolina Museum of Art, Raleigh. Gift of Mr. & Mrs. Gordon Hanes
J
I
E
196 Chapter 4 Congruent Triangles
D
B
C
G
ALGEBRA For Exercises 29 and 30, use the following information.
nQRS > nGHJ, RS 5 12, QR 5 10, QS 5 6, and HJ 5 2x 2 4.
29. Draw and label a figure to show the congruent triangles.
30. Find x.
ALGEBRA For Exercises 31 and 32, use the following information.
nJKL > nDEF, m/J 5 36, m/E 5 64, and m/F 5 3x 1 52.
31. Draw and label a figure to show the congruent triangles.
32. Find x.
33. PROOF The statements below can be used to prove that congruence of triangles
is symmetric. Use the statements to construct a correct flow proof. Provide the
reasons for each statement.
Given: nRST > nXYZ
Prove: nXYZ > nRST
Flow Proof:
R
S
/R > /X, /S >
/Y, /T > /Z,
XY > RS, YZ >
ST, XZ > RT
RS > XY, ST >
YZ, RT > XZ
34. PROOF
Y
T
/X > /R, /Y >
/S, /Z > /T,
?
X
Z
nRST > nXYZ
nXYZ > nRST
?
?
?
Copy the flow proof and provide the reasons for each statement.
B>C
CB
DC
Given: A
ww
wD
w, A
wD
w>w
w, A
wD
w'w
w,
Aw
B'w
BC
BC
CD
w
w, A
wD
w\w
w, A
wB
w\w
w
Prove: nACD > nCAB
B
C
4
2
3
1
A
D
Flow Proof:
AB > CD
AD > CB
AC > CA
AD ' DC
AB ' BC
AD || BC
AB || CD
a. ?
b. ?
c. ?
d. ?
f. ?
i. ?
k. ?
/D is a rt. /.
/B is a rt. /.
/1 > /4
/2 > /3
e. ?
g. ?
j. ?
l. ?
/D > /B
h. ?
n ACD > nCAB
m. ?
35. PROOF Write a flow proof to prove Congruence of triangles is reflexive.
(Theorem 4.4)
36. CRITICAL THINKING nRST is isosceles with RS = RT,
M, N, and P are midpoints of their sides, /S > /MPS,
and N
wP
w>M
wP
w. What else do you need to know to prove
that nSMP > nTNP?
www.geometryonline.com/self_check_quiz
R
M
S
N
P
T
Lesson 4-3 Congruent Triangles 197
37. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson.
Why are triangles used in bridges?
Include the following in your answer:
• whether the shape of the triangle matters, and
• whether the triangles appear congruent.
Standardized
Test Practice
38. Determine which statement is true given nABC > nXYZ.
A
B
ZX
wC
w>w
w
39. ALGEBRA
A
B
A
XZ
wC
w>w
w
A
B>w
YZ
ww
w
C
D
cannot be
determined
D
185
Ïw
Find the length of w
DF
w if D(25, 4) and F(3, 27).
Ïw5
B
13
Ïw
57
Ïw
C
Maintain Your Skills
Mixed Review
Find x. (Lesson 4-2)
40.
41.
40˚
x˚
42.
x˚
x˚
30˚
115˚
x˚
100˚
42˚
Find x and the measure of each side of the triangle. (Lesson 4-1)
43. nBCD is isosceles with B
wC
w>C
wD
w, BC 5 2x 1 4, BD 5 x 1 2, and CD 5 10.
44. Triangle HKT is equilateral with HK 5 x 1 7 and HT 5 4x 2 8.
Write an equation in slope-intercept form for the line that satisfies the given
conditions. (Lesson 3-4)
3
45. contains (0, 3) and (4, 23)
46. m 5 }}, y-intercept 5 8
4
47. parallel to y 5 24x 1 1;
48. m 5 24, contains (23, 2)
contains (23, 1)
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Find the distance between each pair of points.
(To review the Distance Formula, see Lesson 1-4.)
49. (21, 7), (1, 6)
50. (8, 2), (4, 22)
51. (3, 5), (5, 2)
P ractice Quiz 1
Lessons 4-1 through 4-3
1. Identify the isosceles triangles in the figure, if w
Fw
H and w
Dw
G are congruent
perpendicular bisectors. (Lesson 4-1)
ALGEBRA nABC is equilateral with AB 5 2x, BC 5 4x 2 7,
and AC 5 x 1 3.5. (Lesson 4-1)
2. Find x.
3. Find the measure of each side.
4. Find the measure of each numbered angle.
(Lesson 4-2)
F
D
J
H
1 3
50˚
2
21˚
70˚
5. If nMNP > nJKL, name the corresponding congruent angles and sides. (Lesson 4-3)
198 Chapter 4 Congruent Triangles
G

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