Download Congruent Triangles

Congruent Triangles
Why?
Then
You identified triangles
with congruent sides.
The flag of the United Kingdom is shown below. Consider the four large
triangles appearing on the top and the bottom of the flag.
(Lesson 9-3)
Now
Identify corresponding
parts of congruent
triangles.
Identify congruent
triangles.
New Vocabulary
congruent
corresponding parts
Math Online
glencoe.com
Extra Examples
Personal Tutor
Self-Check Quiz
Homework Help
a. Measure the sides of the four triangles. What is true about the lengths?
b. Use a protractor to measure the angles of the four triangles. What is true
about the measures?
c. MAKE A CONJECTURE Suppose the triangles were cut out and laid on top
of one another so that the parts with the same measures were matched up.
What is true about the triangles?
Corresponding Parts of Congruent Triangles Figures that have the same size
and shape are congruent. In the figure below, triangle ABC is congruent to
triangle DEF.
Arcs are used to
indicate which angles
are congruent.
"
$
%
#
Tick marks are used
to indicate which
sides are congruent.
&
'
The parts of congruent triangles that match or correspond, are called
corresponding parts.
Key Concept
Words
For Your
Corresponding Parts of
Congruent Triangles
If two triangles are congruent, their corresponding sides
are congruent and their corresponding angles are
congruent.
Model
"
$
#
%
&
'
ABC DEF
Symbols
598 Chapter 11 Distance and Angle
Congruent Angles: ∠ A ∠D, ∠B ∠E, ∠C ∠F
−− −− −− −− −− −−
Congruent Sides: AB DE, BC EF, CA FD
In a congruence statement, the letters are written so that corresponding vertices
appear in the same order.
StudyTip
Congruence Statements
You can write the
following congruence
statements for the
congruent triangles
at the right.
RSQ UVT
SQR VTU
SRQ VUT
RQS UTV
QSR TVU
QRS TUV
3
6
2
4
5
7
Vertex Q corresponds to vertex T.
Vertex R corresponds to vertex U.
Vertex S corresponds to vertex V.
QRS TUV
Name Corresponding Parts
EXAMPLE 1
a. Name the corresponding parts in the congruent triangles below. Then
? .
complete the congruence statement JKL +
.
,
/
-
1
Use the matching arcs and tick marks to identify the corresponding parts.
Corresponding angles:
Corresponding sides:
∠J ∠M, ∠K ∠N, ∠L ∠P
−− −−− −− −− −− −−−
JK MN, KL NP, LJ PM
The congruence statement is JKL MNP.
b. If ABC HGF, name the corresponding parts. Then complete
? .
the congruence statement BAC StudyTip
Draw a Figure When
given only a congruence
statement, it is helpful to
draw the two triangles in
the same position. Then
it will be easier to
visualize and confirm the
corresponding angles
and sides.
)
"
(
#
Use the order of the vertices in the congruence statement ABC HGF
to identify the corresponding parts.
Corresponding angles:
Corresponding sides:
The congruence statement is BAC GHF.
✓ Check Your Progress
Name the corresponding parts in each pair of congruent triangles. Then
complete the congruence statement.
1A.
$
∠A ∠H, ∠B ∠G, ∠C ∠F
−− −−− −− −− −− −−
AB HG, BC GF, CA FH
'
1B.
3
'
TUV CDE
VTU %
&
DEF 2
?
1
?
Personal Tutor glencoe.com
Lesson 11-2 Congruent Triangles
599
You can use corresponding parts to find the measures of angles and sides
in a figure that is congruent to a figure with known measures.
EXAMPLE 2
Find Missing Measures
SIGNS In the signs below, RST VWX.
S
R
W
T
X
V
a. If m∠W = 60°, what is the measure of ∠S?
∠W and ∠S are corresponding angles, so they are congruent.
m∠S = 60°
Congruent angles have equal measures.
−−
−−−
b. The length of ST is 30 inches. What is the length of WX?
−−
−−−
ST and WX are corresponding sides, so they are congruent.
WX = 30 inches
Congruent sides have equal lengths.
✓ Check Your Progress
2. QUILTING In the quilt design
shown, ABC ADE. What
is the measure of ∠BCA? What is
the length of AD?
18 cm
#
"
%
3 cm
59°
59
$
&
Personal Tutor glencoe.com
Identify Congruent Triangles In congruent triangles, all corresponding angles
are congruent and all corresponding sides are congruent.
StudyTip
Identify Congruent
Parts To write a
congruence statement,
note which angles are
congruent and which
segments have the same
length. Then write
corresponding vertices
in the same order.
EXAMPLE 3
Identify Congruent Triangles
(
Determine whether the triangles shown
at the right are congruent. If so, name the
corresponding parts and write a
congruence statement.
6 cm
2
3
9.5 cm 9.5 cm
8 cm
'
)
The arcs indicate that ∠G ∠P, ∠H ∠Q,
6 cm
and ∠F ∠R.
−−− −− −− −−−
−− −−
The side measures indicate that GH PQ, HF QR, and FG RP.
8 cm
1
Since all pairs of corresponding angles and sides are congruent, the triangles
are congruent. One congruence statement is GHF PQR.
*
✓ Check Your Progress
3. Determine whether the triangles
shown are congruent. If so, name
the corresponding parts and write a
congruence statement.
8 in.
11 in.
)
3 in.
+
6 in.
(
4 in.
,
14 in.
-
Personal Tutor glencoe.com
600 Chapter 11 Distance and Angle
✓ Check Your Understanding
Example 1
p. 599
Name the corresponding parts in each pair of congruent triangles.
Then complete the congruence statement.
1.
9
.
:
-
2. RST GJK
?
STR /
;
YZX ?
+
,
Example 2
p. 600
3. KITES In the umbrella kite shown at the right,
JLK NLM.
-
a. If m∠JKL = 67.5°, what is m∠NML?
.
b. If NL = 14 inches and MN = 12 inches,
what is JK?
Example 3
p. 600
/
Determine whether the triangles shown are congruent. If so, name the
corresponding parts and write a congruence statement.
4.
5.
%
"
12.5 ft
$
/
'
18 ft
15 ft
+
#
'
$
p. 599
15 ft
8
&
= Step-by-Step Solutions begin on page R11.
Extra Practice begins on page 810.
Practice and Problem Solving
Example 1
2
Name the corresponding parts in each pair of congruent triangles.
Then complete the congruence statement.
6. "
(
#
'
?
5
?
:
+
STZ 8. KLM WXY
LKM 4
;
$
GJF 7
8
?
9. NPQ DEF
EFD ?
Lesson 11-2 Congruent Triangles
601
Example 2
p. 600
10. STAINED GLASS In the stained glass window at the
right, ABC ADC.
"
a. If BC = 26 centimeters, what is DC?
b. If m∠ADC = 90°, and m∠DCA = 45°, what
is m∠BAC?
#
%
11. ALGEBRA Triangle STU is congruent to triangle
MNP. In the figures, TU = 15.4 millimeters,
US = 13 millimeters, and ST = 18.1 millimeters.
If MN = 2x + 1, what is the value of x?
Example 3
p. 600
$
Determine whether the triangles shown are congruent. If so, name the
corresponding parts and write a congruence statement.
12.
13.
1
;
1
42°
/
4
8 m 65°
8
2
78°
.
7
14. '
21°
6 cm
9 cm
3
(
4 cm
104°
4 cm 6 cm
32° )
4
15.
13 m
13 m
37°
12 m
-
12 m
63°
0
,
75°
9m
2
3
1
8 cm
+
29°
2
3
2
16. ART The structure shown at the right, Moondog, created
by Tony Smith, is located at the Hirshhorn Museum
and Sculpture Garden in Washington, D.C. If
ABC DFG, name all corresponding sides.
O
M
P
N
The average weight of an
individual block of stone
that makes up the Khafre
pyramid in Egypt is 2.5
tons. Some of the outside
stones weigh up to 7 tons.
A
B
D
C
F
G
17 SOCIAL STUDIES The Khafre pyramid shown at the
left is composed of triangles. If MNO NPO,
B m∠M = 60°, and m∠OPN = 50°, find m∠O.
18. GEOMETRY Hexagon ABCDEF has six congruent sides.
−− −−
−−
Segments CA , CF, and CE are drawn on the hexagon,
forming four triangles. Make a conjecture about which
triangles are congruent. Test your conjecture by measuring
the sides and angles of the triangles.
Source: PBS
19. In the figure below, FGI LKM. Find the value of x.
'
,
5 cm
2x + 4 cm
*
602 Chapter 11 Distance and Angle
.
10 cm
(
-
#
$
%
"
'
&
20. CONSTRUCTION The roof at the right
contains two congruent triangles.
a. Write a congruence statement
involvingthe triangles. Then write
a congruence statement involving
a pair of corresponding sides.
:
9
;
8
14 ft
b. What is the distance from the right side
of the roof at the base to the center? Name the side.
−−−
−−−
−−−
c. If the length of XW is 4 feet longer than WY, what is the length of WY ?
21. BUTTERFLIES Butterfly wings are triangular in
shape. Using the art of the butterfly as a model,
draw two sets of congruent triangles, label the
vertices, and write a congruence statement
for each.
Biological Scientists
Biological scientists study
the behavior, genetics, and
diseases of living things.
Since the job requires
collecting and analyzing
data, knowledge of
arithmetic, algebra and
statistics are important.
A four-year degree in a
biological field is the
minimum level of
education required for
this career.
C
ALGEBRA Find the value of x for each pair of congruent triangles.
22.
32
)
,
23
7
-
.
15
;
34
28
1
+
3x + 5
9
H.O.T. Problems
4x - 9
19
/
8
Use Higher-Order Thinking Skills
24. OPEN ENDED Find a real-world example of congruent triangles. Explain how
you know the triangles are congruent.
25. FIND THE ERROR Jade and Fernando are writing a
congruence statement for the congruent triangles
at the right. Is either of them correct? Justify
your reasoning.
4
5
9
:
Jade
Fernando
∠YXZ ∠STR
−−
−−
ST ZY
;
3
26. CHALLENGE Determine whether each statement is true or false. If true, explain
your reasoning. If false, give a counterexample.
a. Triangles EFG and KLM are congruent. So, their perimeters are equal.
b. Triangles WXY and HJK are congruent. So, their areas are equal.
c. The perimeter of ABC is 24 millimeters and the perimeter of RST is
24 millimeters. So, triangles ABC and RST are congruent.
27. CHALLENGE Describe the scale factor of two congruent triangles.
28. WRITING IN MATH Describe structures that are built using congruent
triangles. Explain why you think congruent triangles are used.
Lesson 11-2 Congruent Triangles
603
Standardized Test Practice
29. In the road sign below, ABC DCB,
AC is 2.5 meters long, BC is 1 meter long,
and AB is 2.7 meters long. What is the length
−−
of BD?
#
25 North
64
%
31. Which of the following statement is not true
if FGH LMN?
−− −−
A FH LN
−−− −−−
B GH MN
C ∠G ∠M
"UMBOUB
"JSQPSU
&YJU
D ∠F ∠N
"
$
A 1 meter
B 2.5 meters
C 2.7 meters
D 2 meters
30. Guy–wires connected to a telephone pole
create two congruent triangles PQR
−−
and SQR. Find the length of QS.
F
G
H
J
32. EXTENDED RESPONSE Determine whether the
triangles shown below are congruent. If so,
name the corresponding parts and write a
congruence statement.
3
12 ft
24 ft
48 ft
65 ft
%
"
60°
60°
30°
30°
1
2
4
$
#
'
&
24 ft
Spiral Review
33. Angles R and S are complementary. Find m∠R if m∠S = 65.7° (Lesson 11-1)
34. In a 45°-45°-90° triangle, a leg is 4 centimeters long. Find the length of the
hypotenuse to the nearest tenth. (Lesson 10-6)
35. BIRDS A mockingbird uses about 5 -4 Joules of energy to sing a song. Write
the amount of energy the bird uses as an expression using a positive
exponent. (Lesson 9-4)
36. CAR RENTAL The costs for renting a car from Able Car
Rental and from Baker Car Rental are shown in the table.
For what mileage does Baker have the better deal? Use
the inequality 30 + 0.05x > 20 + 0.10x. Explain why this
inequality works. (Lesson 5-5)
Rental Car Costs
Cost per Day Cost per Mile
Able Car
Rental
Baker Car
Rental
$30
$0.05
$20
$0.10
Skills Review
Graph each ordered pair on a coordinate plane. (Lesson 2-6)
37. F (4, 5)
38. G (2, -2)
604 Chapter 11 Distance and Angle
39. C (-1.4)
40. A (-3, -2)