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Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
1.
YZX
≅ ?
ANSWER:
∠L X, M Y, N Z,
≅∠ ∠ ≅∠ ∠ ≅∠
2.
RST GJK
≅
STR
≅ ?
ANSWER:
R G, S J, T K,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
3.
KITES In the umbrella kite shown, JLK
≅
NLM.
a. If m JKL m NML?
∠ = 67.5°, what is ∠
b. If NL = 14 inches and MN = 12 inches, what is
JK?
ANSWER:
a.
67.5°
11-2 Congruent Triangles b. 12 in.
Name the corresponding parts in each pair of Determine whether the triangles shown are
congruent triangles. Then complete the
congruent. If so, name the corresponding parts
congruence statement.
and write a congruence statement.
1.
4.
ANSWER:
YZX
≅ ?
yes; A D, B E, C F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER: Sample answer:
ABC DEF
Δ ≅Δ
∠L X, M Y, N Z,
≅∠ ∠ ≅∠ ∠ ≅∠
5.
2.
RST GJK
≅
STR
≅ ?
ANSWER:
ANSWER:
R G, S J, T K,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
no
Name the corresponding parts in each pair of
3. KITES In the umbrella kite shown, JLK
≅ congruent triangles. Then complete the
NLM.
congruence statement.
6.
GJF ?
≅
a. If m JKL m NML? ANSWER:
∠ = 67.5°, what is ∠
A F, B G, C J,
b. If NL = 14 inches and MN = 12 inches, what is ∠ ≅∠ ∠ ≅∠ ∠ ≅∠
JK?
ANSWER:
a. 7.
67.5°
b. 12 in.
Determine whether the triangles shown are
congruent. If so, name the corresponding parts
and write a congruence statement.
4. STZ ?
≅
eSolutions Manual - Powered by Cognero ANSWER: Page1
∠S ∠Y,∠STZ ∠YTW,∠Z ∠W,
≅ ≅ ≅
8.
KLM WXY
≅
ANSWER:
LKM ?
≅
yes; A D, B E, C F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
Sample answer:
K W, L X, M Y,
ABC DEF ∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Δ ≅Δ
5.
9. NPQ DEF
≅
EFD ?
≅
ANSWER:
N D, P E, Q F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
no
Name the corresponding parts in each pair of 10. STAINED GLASS In the stained glass window
shown, ABC ADC.
congruent triangles. Then complete the Δ ≅Δ
congruence statement.
6.
GJF ?
≅
ANSWER: a. If BC = 26 centimeters, what is DC?
A F, B G, C J,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠ b.
If m ADC = 90°, and m DCA = 45°, what is
∠ ∠
m BAC?
∠
ANSWER:
7. a. 26 cm
b.
45°
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 +
STZ ? 1, what is the value of x?
≅
ANSWER:
ANSWER: 8.55 mm
∠S ∠Y,∠STZ ∠YTW,∠Z ∠W,
≅ ≅ ≅
Determine whether the triangles shown are
congruent. If so, name the corresponding parts
8. and write a congruence statement.
KLM WXY
≅
12.
LKM ?
≅
ANSWER:
K W, L X, M Y,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
9. NPQ DEF
≅
EFD ?
≅
ANSWER:
ANSWER:
yes; P V, W Q, Z S,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
N D, P E, Q F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
ΔZWP ΔSQV
≅
10. STAINED GLASS In the stained glass window
shown, ABC ADC.
Δ ≅Δ
13.
ANSWER:
no
a. If BC = 26 centimeters, what is DC? 14.
b.
If m ADC = 90°, and m DCA = 45°, what is
∠ ∠
m BAC?
∠
ANSWER:
a. 26 cm
b.
45°
ANSWER:
11. ALGEBRA Triangle STU is congruent to triangle no
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? 15.
ANSWER:
8.55 mm
Determine whether the triangles shown are
congruent. If so, name the corresponding parts
ANSWER:
and write a congruence statement.
yes; J P, L R, K Q,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
12.
Sample answer:
JLK PRQ
Δ ≅Δ
16.
If ABC DFG, name all corresponding sides.
Δ ≅Δ
ANSWER:
ANSWER:
17. SOCIAL STUDIES The Khafre pyramid shown
below is composed of triangles.
yes; P V, W Q, Z S,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠ (See page 636 of your text.)
Sample answer:
If MNO NPO, m M m OPN=
Δ ≅Δ ∠ = 60°, and ∠
ΔZWP ΔSQV
≅ m O.
50°, find ∠
ANSWER:
13.
70°
18.
GEOMETRY Hexagon ABCDEF has six
congruent sides. Segments , , and 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.
ANSWER:
no
14.
ANSWER:
Triangles CBA and CDE are congruent. Triangles
CAF and CEF are congruent.
ANSWER:
19. In the figure below, FGI LKM. Find the value
no Δ ≅Δ
of x. Round to the nearest tenth.
15.
ANSWER:
3.6 cm
ANSWER:
20.
CONSTRUCTION The roof contains two
yes; J P, L R, K Q,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠ congruent triangles.
Sample answer:
JLK PRQ
Δ ≅Δ
16.
If ABC DFG, name all corresponding sides.
Δ ≅Δ
ANSWER:
a. Write a congruence statement involving the
triangles. Then write a congruence statement
17.
SOCIAL STUDIES The Khafre pyramid shown involving a pair of corresponding sides.
below is composed of triangles. b. What is the distance from the right side of the
(See page 636 of your text.)
roof at the base to the center? Name the side.
If MNO NPO, m M m OPN=
Δ ≅Δ ∠ = 60°, and ∠
c. If the length of ,
m O. is 4 feet longer than
50°, find ∠ what is the length of ?
ANSWER:
ANSWER:
70°
a. Δ
Sample answer: XYW ΔZYW;
≅ ≅
18. GEOMETRY Hexagon ABCDEF has six b. 7 ft;
congruent sides. Segments , , and are
drawn on the hexagon, forming four triangles. Make c. 3 ft
a conjecture about which triangles are congruent.
Test your conjecture by measuring the sides and 21. BUTTERFLIES Butterfly wings are triangular in
angles of the triangles. 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.
ANSWER:
Triangles CBA and CDE are congruent. Triangles
CAF and CEF are congruent. ANSWER:
Sample answer:
19.
In the figure below, ΔFGI ΔLKM. Find the value
≅
of x. Round to the nearest tenth.
ANSWER:
3.6 cm
GHI JKL, MNO PQR
≅ ≅
20.
CONSTRUCTION The roof contains two x for each pair of
congruent triangles. ALGEBRA Find the value of
congruent triangles.
22.
a. Write a congruence statement involving the
triangles. Then write a congruence statement
involving a pair of corresponding sides.
b. What is the distance from the right side of the ANSWER:
9
roof at the base to the center? Name the side.
c. If the length of ,
is 4 feet longer than
what is the length of ? 23.
ANSWER:
a. Δ
Sample answer: XYW ΔZYW;
≅ ≅
b. 7 ft;
c. 3 ft
ANSWER:
21. BUTTERFLIES Butterfly wings are triangular in 7
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. 24. OPEN ENDED Find a real-world example of
congruent triangles. Explain how you know the
triangles are congruent.
ANSWER:
See students’ work.
25.
ERROR ANALYSIS Jade and Fernando are
writing a congruence statement for the congruent
triangles shown. Is either of them correct? Justify
ANSWER: your reasoning.
Sample answer:
GHI JKL, MNO PQR
≅ ≅
x for each pair of
ALGEBRA Find the value of
congruent triangles.
22.
ANSWER:
No; corresponding vertices must be written in the
same order, so YXZ SRT, not STR and
∠ ≅∠ ∠
, not .
≅
26.
CHALLENGE Determine whether each statement
is true or false. If true, explain your reasoning. If
ANSWER:
9 false, give a counterexample.
a. Triangles EFG and KLM are congruent. So, their
perimeters are equal.
23. b. Triangles WXY and HJK are congruent. So, their
areas are equal.
c. The perimeter of ΔABC is 24 millimeters and the
perimeter of ΔRST is 24 millimeters. So, triangles
ABC and RST are congruent.
ANSWER:
a. True; If the triangles are congruent, then the
ANSWER: corresponding sides have equal lengths. Therefore,
7
the sums of the lengths, or the perimeters, are equal.
b. True; If the triangles are congruent, then the
24. OPEN ENDED Find a real-world example of corresponding sides have equal lengths and the
congruent triangles. Explain how you know the corresponding bases and heights are equal.
triangles are congruent.
Therefore, the products base height, or the
• •
ANSWER:
See students work.
’
areas, are equal.
c. False; Sample answer: the sides of ABC
Δ
25. ERROR ANALYSIS Jade and Fernando are measure 8 mm, 8mm, and 8 mm. The sides of RST
writing a congruence statement for the congruent Δ
triangles shown. Is either of them correct? Justify measure 6 mm, 8 mm, and 10 mm.
your reasoning.
27. CHALLENGE Describe the scale factor of two
congruent triangles.
ANSWER:
The scale factor is 1:1 or 1.
28.
WRITING IN MATH Describe structures that
are built using congruent triangles. Explain why you
think congruent triangles are used.
ANSWER:
Sample answer: Congruent triangles are used in
bridges and roofs. Congruent triangles help balance
and support the structures.
29. In the road sign, ABC DCB, AC is 2.5 meters
∆ ∆
ANSWER: long. What is the length of ?
No; corresponding vertices must be written in the
same order, so YXZ SRT, not STR and
∠ ≅∠ ∠
, not .
≅
26.
CHALLENGE Determine whether each statement
is true or false. If true, explain your reasoning. If
false, give a counterexample. A
a. Triangles EFG and KLM are congruent. So, their 1 meter
B
perimeters are equal. 2.5 meters
C
b. Triangles WXY and HJK are congruent. So, their 2.7 meters
D
2 meters
areas are equal.
ANSWER:
c. The perimeter of ABC is 24 millimeters and the
Δ
perimeter of RST is 24 millimeters. So, triangles B
Δ
ABC and RST are congruent.
30. Guy wires connected to a telephone pole create two
ANSWER: congruent triangles PQR and SQR. Find the
∆ ∆
a. True; If the triangles are congruent, then the length of .
corresponding sides have equal lengths. Therefore,
the sums of the lengths, or the perimeters, are equal.
b. True; If the triangles are congruent, then the
corresponding sides have equal lengths and the
corresponding bases and heights are equal.
Therefore, the products base height, or the
• •
areas, are equal.
c. False; Sample answer: the sides of ABC
Δ F
measure 8 mm, 8mm, and 8 mm. The sides of RST 12 ft
Δ G
measure 6 mm, 8 mm, and 10 mm. 24 ft
H
48 ft
J
65 ft
27. CHALLENGE Describe the scale factor of two
congruent triangles.
ANSWER:
ANSWER: F
The scale factor is 1:1 or 1.
31. Which of the following statements is not true if
FGH LMN?
28. ∆ ∆
WRITING IN MATH Describe structures that
are built using congruent triangles. Explain why you A
think congruent triangles are used.
B
ANSWER: C G M
Sample answer: Congruent triangles are used in ∠ ∠
D
F N
bridges and roofs. Congruent triangles help balance ∠ ∠
and support the structures.
ANSWER:
D
29. In the road sign, ABC DCB, AC is 2.5 meters
∆ ∆
32. EXTENDED RESPONSE
long. What is the length of ? Determine whether the
triangles shown are congruent. If so, name the
corresponding parts and write a congruence
statement.
A
1 meter
B
2.5 meters
C
2.7 meters
D ANSWER:
2 meters
yes; A D, B E, C F,
∠ ∠ ∠ ∠ ∠ ∠
ANSWER: , , ; Sample
B
answer: ABC DEF
∆ ∆
30. Guy wires connected to a telephone pole create two
33. Angles R and S are complementary. Find m R if
congruent triangles PQR and SQR. Find the ∠
∆ ∆ m S
length of . ∠ = 65.7°.
ANSWER:
24.3°
34. - -
In a 45° 45° 90° triangle, a leg is 4 centimeters long.
Find the length of the hypotenuse to the nearest
tenth.
ANSWER:
F 5.7 cm
12 ft
G –4
24 ft
H 35. BIRDS A mockingbird uses about 5 Joules of
48 ft energy to sing a song. Write the amount of energy
J
65 ft the bird uses as an expression using a positive
exponent.
ANSWER:
F ANSWER:
31. Which of the following statements is not true if
FGH LMN?
∆ ∆
36.
A CAR RENTAL The costs for renting a car from
Able Car Rental and from Baker Car Rental are
B
shown in the table. For what mileage does Baker
C G M
∠ ∠ have the better deal? Use the inequality 30 + 0.05x >
D
F N 20 + 0.10x. Explain why this inequality works.
∠ ∠
ANSWER:
D
32. EXTENDED RESPONSE
Determine whether the
triangles shown are congruent. If so, name the
corresponding parts and write a congruence
statement.
ANSWER:
x < 200; Sample answer: The inequality finds at what
mileage Able s charge is greater than Baker s
’ ’
charge.
ANSWER:
Graph each ordered pair on a coordinate plane.
yes; A D, B E, C F,
∠ ∠ ∠ ∠ ∠ ∠
, , ; Sample 37. F (4, 5)
answer: ABC DEF ANSWER:
∆ ∆
33.
Angles R and S are complementary. Find m R if
∠
m S= 65.7°.
∠
ANSWER:
24.3°
34. - -
In a 45° 45° 90° triangle, a leg is 4 centimeters long.
Find the length of the hypotenuse to the nearest
tenth.
ANSWER:
5.7 cm
38. G (2, 2)
–4 –
35. BIRDS A mockingbird uses about 5 Joules of
energy to sing a song. Write the amount of energy ANSWER:
the bird uses as an expression using a positive
exponent.
ANSWER:
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.
39.
C(–1, 4)
ANSWER:
ANSWER:
x < 200; Sample answer: The inequality finds at what
mileage Able s charge is greater than Baker s
’ ’
charge.
40.
Graph each ordered pair on a coordinate plane. A (–3, –2)
37.
F(4, 5)
ANSWER:
ANSWER:
38. G (2, 2)
–
ANSWER:
39.
C(–1, 4)
ANSWER:
40.
A (–3, –2)
ANSWER:
Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
1.
YZX
≅ ?
ANSWER:
∠L X, M Y, N Z,
≅∠ ∠ ≅∠ ∠ ≅∠
2.
RST GJK
≅
STR
≅ ?
ANSWER:
R G, S J, T K,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
3.
KITES In the umbrella kite shown, JLK
≅
NLM.
a. If m JKL m NML?
∠ = 67.5°, what is ∠
b. If NL = 14 inches and MN = 12 inches, what is
JK?
ANSWER:
a.
67.5°
b. 12 in.
Determine whether the triangles shown are
congruent. If so, name the corresponding parts
and write a congruence statement.
4.
Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
1.
ANSWER:
yes; A D, B E, C F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
ABC DEF
Δ ≅Δ
YZX
≅ ?
5.
ANSWER:
∠L X, M Y, N Z,
≅∠ ∠ ≅∠ ∠ ≅∠
2.
RST GJK
≅
ANSWER:
STR ?
≅ no
ANSWER: Name the corresponding parts in each pair of
R G, S J, T K,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠ congruent triangles. Then complete the
congruence statement.
6.
3.
KITES In the umbrella kite shown, JLK
≅
NLM.
GJF ?
≅
ANSWER:
A F, B G, C J,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
a. If m JKL m NML?
∠ = 67.5°, what is ∠
b. If NL = 14 inches and MN = 12 inches, what is 7.
JK?
ANSWER:
a.
67.5°
b. 12 in.
STZ ?
Determine whether the triangles shown are ≅
congruent. If so, name the corresponding parts
ANSWER:
and write a congruence statement.
∠S ∠Y,∠STZ ∠YTW,∠Z ∠W,
≅ ≅ ≅
4.
8.
KLM WXY
≅
LKM ?
≅
ANSWER:
K W, L X, M Y,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
yes; A D, B E, C F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
9. NPQ DEF
≅
ABC DEF
Δ ≅Δ EFD ?
≅
ANSWER:
5.
N D, P E, Q F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
10. STAINED GLASS In the stained glass window
shown, ABC ADC.
Δ ≅Δ
ANSWER:
no
Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
6.
a. If BC = 26 centimeters, what is DC?
b.
If m ADC = 90°, and m DCA = 45°, what is
∠ ∠
GJF ?
≅
m BAC?
∠
ANSWER:
ANSWER:
A F, B G, C J,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠ a. 26 cm
11-2 Congruent Triangles b.
45°
7. 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?
ANSWER:
8.55 mm
STZ ?
≅ Determine whether the triangles shown are
congruent. If so, name the corresponding parts
ANSWER:
and write a congruence statement.
∠S ∠Y,∠STZ ∠YTW,∠Z ∠W,
≅ ≅ ≅
12.
8.
KLM WXY
≅
LKM ?
≅
ANSWER:
K W, L X, M Y,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
9.
NPQ DEF yes; P V, W Q, Z S,
≅ ∠ ≅∠ ∠ ≅∠ ∠ ≅∠
EFD ?
≅
Sample answer:
ΔZWP ΔSQV
≅
ANSWER:
N D, P E, Q F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
13.
10. STAINED GLASS In the stained glass window
shown, ABC ADC.
Δ ≅Δ
ANSWER:
no
14.
a. If BC = 26 centimeters, what is DC?
b.
If m ADC = 90°, and m DCA = 45°, what is
∠ ∠
m BAC?
∠
ANSWER:
ANSWER: no
a. 26 cm
b.
45°
15.
11.
ALGEBRA Triangle STU is congruent to triangle
MNP. In the figures, TU = 15.4 millimeters, US = 13
eSolutions Manual - Powered by Cognero Page2
millimeters, and ST = 18.1 millimeters. If MN = 2x +
1, what is the value of x?
ANSWER:
8.55 mm ANSWER:
yes; J P, L R, K Q,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Determine whether the triangles shown are Sample answer:
congruent. If so, name the corresponding parts JLK PRQ
Δ ≅Δ
and write a congruence statement.
12.
16. If ABC DFG, name all corresponding sides.
Δ ≅Δ
ANSWER:
17.
SOCIAL STUDIES The Khafre pyramid shown
below is composed of triangles.
(See page 636 of your text.)
If MNO NPO, m M m OPN=
Δ ≅Δ ∠ = 60°, and ∠
ANSWER: m O.
50°, find ∠
yes; P V, W Q, Z S,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
Sample answer:
70°
ΔZWP ΔSQV
≅
18.
GEOMETRY Hexagon ABCDEF has six
13. congruent sides. Segments , , and
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.
ANSWER:
no
14.
ANSWER:
Triangles CBA and CDE are congruent. Triangles
CAF and CEF are congruent.
19.
In the figure below, ΔFGI ΔLKM. Find the value
≅
of x. Round to the nearest tenth.
ANSWER:
no
15.
ANSWER:
3.6 cm
20.
CONSTRUCTION The roof contains two
congruent triangles.
ANSWER:
yes; J P, L R, K Q,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
JLK PRQ
Δ ≅Δ
a. Write a congruence statement involving the
16. If ABC DFG, name all corresponding sides.
Δ ≅Δ triangles. Then write a congruence statement
involving a pair of corresponding sides.
ANSWER: 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 ,
is 4 feet longer than
17. SOCIAL STUDIES The Khafre pyramid shown
below is composed of triangles. what is the length of ?
(See page 636 of your text.)
ANSWER:
If MNO NPO, m M m OPN=
Δ ≅Δ ∠ = 60°, and ∠
50°, find m O. a. Δ
∠ Sample answer: XYW ΔZYW;
≅ ≅
b. 7 ft;
ANSWER:
c. 3 ft
70°
21.
BUTTERFLIES Butterfly wings are triangular in
18. GEOMETRY Hexagon ABCDEF has six shape. Using the art of the butterfly as a model,
congruent sides. Segments , , and are draw two sets of congruent triangles, label the
drawn on the hexagon, forming four triangles. Make vertices, and write a congruence statement for each.
a conjecture about which triangles are congruent.
Test your conjecture by measuring the sides and
angles of the triangles.
ANSWER:
Sample answer:
ANSWER:
Triangles CBA and CDE are congruent. Triangles
CAF and CEF are congruent.
19.
In the figure below, ΔFGI ΔLKM. Find the value
≅
of x. Round to the nearest tenth.
GHI JKL, MNO PQR
≅ ≅
x for each pair of
ALGEBRA Find the value of
ANSWER:
3.6 cm congruent triangles.
22.
20.
CONSTRUCTION The roof contains two
congruent triangles.
ANSWER:
9
a. Write a congruence statement involving the
triangles. Then write a congruence statement
involving a pair of corresponding sides.
b. What is the distance from the right side of the 23.
roof at the base to the center? Name the side.
c. If the length of ,
is 4 feet longer than
what is the length of ?
ANSWER:
a. Δ ANSWER:
Sample answer: XYW ΔZYW;
≅ ≅
7
b. 7 ft;
c. 3 ft
24. OPEN ENDED Find a real-world example of
congruent triangles. Explain how you know the
21. BUTTERFLIES Butterfly wings are triangular in triangles are congruent.
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. ANSWER:
See students’ work.
25.
ERROR ANALYSIS Jade and Fernando are
writing a congruence statement for the congruent
triangles shown. Is either of them correct? Justify
your reasoning.
ANSWER:
Sample answer:
GHI JKL, MNO PQR
≅ ≅
x for each pair of ANSWER:
ALGEBRA Find the value of No; corresponding vertices must be written in the
congruent triangles. same order, so YXZ SRT, not STR and
∠ ≅∠ ∠
22.
, not .
≅
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.
ANSWER: b. Triangles WXY and HJK are congruent. So, their
9
areas are equal.
c. The perimeter of ΔABC is 24 millimeters and the
perimeter of ΔRST is 24 millimeters. So, triangles
23. ABC and RST are congruent.
ANSWER:
a. True; If the triangles are congruent, then the
corresponding sides have equal lengths. Therefore,
the sums of the lengths, or the perimeters, are equal.
b. True; If the triangles are congruent, then the
ANSWER: corresponding sides have equal lengths and the
7 corresponding bases and heights are equal.
Therefore, the products • base • height, or the
24. OPEN ENDED Find a real-world example of
congruent triangles. Explain how you know the
triangles are congruent. areas, are equal.
c. False; Sample answer: the sides of ΔABC
measure 8 mm, 8mm, and 8 mm. The sides of ΔRST
ANSWER: measure 6 mm, 8 mm, and 10 mm.
See students’ work.
27.
CHALLENGE Describe the scale factor of two
25. ERROR ANALYSIS Jade and Fernando are congruent triangles.
writing a congruence statement for the congruent
triangles shown. Is either of them correct? Justify
your reasoning. ANSWER:
The scale factor is 1:1 or 1.
28.
WRITING IN MATH Describe structures that
are built using congruent triangles. Explain why you
think congruent triangles are used.
ANSWER:
Sample answer: Congruent triangles are used in
bridges and roofs. Congruent triangles help balance
and support the structures.
29. In the road sign, ∆ABC ∆DCB, AC is 2.5 meters
long. What is the length of ?
ANSWER:
No; corresponding vertices must be written in the
same order, so YXZ SRT, not STR and
∠ ≅∠ ∠
, not .
≅ A
1 meter
B
2.5 meters
26. CHALLENGE Determine whether each statement C
is true or false. If true, explain your reasoning. If 2.7 meters
D
false, give a counterexample. 2 meters
a. Triangles EFG and KLM are congruent. So, their
perimeters are equal. ANSWER:
B
b. Triangles WXY and HJK are congruent. So, their
areas are equal.
30. Guy wires connected to a telephone pole create two
c. The perimeter of ΔABC is 24 millimeters and the congruent triangles ∆PQR and ∆SQR. Find the
perimeter of ΔRST is 24 millimeters. So, triangles length of .
ABC and RST are congruent.
ANSWER:
a. True; If the triangles are congruent, then the
corresponding sides have equal lengths. Therefore,
the sums of the lengths, or the perimeters, are equal.
b. True; If the triangles are congruent, then the
corresponding sides have equal lengths and the
corresponding bases and heights are equal.
F
Therefore, the products • base • height, or the 12 ft
G
24 ft
H
areas, are equal. 48 ft
J
c. False; Sample answer: the sides of ΔABC 65 ft
measure 8 mm, 8mm, and 8 mm. The sides of ΔRST
measure 6 mm, 8 mm, and 10 mm. ANSWER:
F
27.
CHALLENGE Describe the scale factor of two
congruent triangles. 31. Which of the following statements is not true if
∆FGH ∆LMN?
ANSWER: A
The scale factor is 1:1 or 1.
B
28. WRITING IN MATH Describe structures that C G M
are built using congruent triangles. Explain why you ∠ ∠
D
F N
think congruent triangles are used. ∠ ∠
ANSWER:
ANSWER: D
Sample answer: Congruent triangles are used in
bridges and roofs. Congruent triangles help balance
32. EXTENDED RESPONSE
and support the structures. Determine whether the
triangles shown are congruent. If so, name the
corresponding parts and write a congruence
29. In the road sign, ∆ABC ∆DCB, AC is 2.5 meters statement.
long. What is the length of ?
ANSWER:
A
1 meter
yes; A D, B E, C F,
B ∠ ∠ ∠ ∠ ∠ ∠
2.5 meters , , ; Sample
C
2.7 meters answer: ∆ABC ∆DEF
D
2 meters
33.
ANSWER: Angles R and S are complementary. Find m R if
∠
m S= 65.7°.
B ∠
ANSWER:
30. Guy wires connected to a telephone pole create two
congruent triangles PQR and SQR. Find the 24.3°
∆ ∆
length of .
34. - -
In a 45° 45° 90° triangle, a leg is 4 centimeters long.
Find the length of the hypotenuse to the nearest
tenth.
ANSWER:
5.7 cm
–4
35. BIRDS A mockingbird uses about 5 Joules of
energy to sing a song. Write the amount of energy
F the bird uses as an expression using a positive
12 ft exponent.
G
24 ft
H
48 ft ANSWER:
J
65 ft
ANSWER:
F
36. CAR RENTAL The costs for renting a car from
Able Car Rental and from Baker Car Rental are
31. Which of the following statements is not true if shown in the table. For what mileage does Baker
FGH LMN?
∆ ∆ have the better deal? Use the inequality 30 + 0.05x >
A 20 + 0.10x. Explain why this inequality works.
B
C G M
∠ ∠
D
F N
∠ ∠
ANSWER:
D
32. EXTENDED RESPONSE
Determine whether the
triangles shown are congruent. If so, name the
corresponding parts and write a congruence
statement. ANSWER:
x < 200; Sample answer: The inequality finds at what
mileage Able s charge is greater than Baker s
’ ’
charge.
Graph each ordered pair on a coordinate plane.
37. F (4, 5)
ANSWER:
ANSWER:
yes; A D, B E, C F,
∠ ∠ ∠ ∠ ∠ ∠
, , ; Sample
answer: ABC DEF
∆ ∆
33.
Angles R and S are complementary. Find m R if
∠
m S= 65.7°.
∠
ANSWER:
24.3°
34. - -
In a 45° 45° 90° triangle, a leg is 4 centimeters long.
Find the length of the hypotenuse to the nearest
tenth.
38. G (2, 2)
–
ANSWER:
ANSWER:
5.7 cm
–4
35. BIRDS A mockingbird uses about 5 Joules of
energy to sing a song. Write the amount of energy
the bird uses as an expression using a positive
exponent.
ANSWER:
36.
CAR RENTAL The costs for renting a car from
Able Car Rental and from Baker Car Rental are
39. C ( 1, 4)
shown in the table. For what mileage does Baker –
have the better deal? Use the inequality 30 + 0.05x > ANSWER:
20 + 0.10x. Explain why this inequality works.
ANSWER:
40.
x < 200; Sample answer: The inequality finds at what A (–3, –2)
mileage Able s charge is greater than Baker s
’ ’
ANSWER:
charge.
Graph each ordered pair on a coordinate plane.
37.
F(4, 5)
ANSWER:
38. G (2, 2)
–
ANSWER:
39.
C(–1, 4)
ANSWER:
40.
A (–3, –2)
ANSWER:
Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
1.
YZX
≅ ?
ANSWER:
∠L X, M Y, N Z,
≅∠ ∠ ≅∠ ∠ ≅∠
2.
RST GJK
≅
STR
≅ ?
ANSWER:
R G, S J, T K,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
3.
KITES In the umbrella kite shown, JLK
≅
NLM.
a. If m JKL m NML?
∠ = 67.5°, what is ∠
b. If NL = 14 inches and MN = 12 inches, what is
JK?
ANSWER:
a.
67.5°
Name the corresponding parts in each pair of b. 12 in.
congruent triangles. Then complete the
congruence statement. Determine whether the triangles shown are
1.
congruent. If so, name the corresponding parts
and write a congruence statement.
4.
YZX
≅ ?
ANSWER:
ANSWER:
yes; A D, B E, C F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
∠L X, M Y, N Z,
≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
ABC DEF
Δ ≅Δ
2.
RST GJK
≅
STR 5.
≅ ?
ANSWER:
R G, S J, T K,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
3. KITES In the umbrella kite shown, JLK
≅ no
NLM.
Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
6.
a. If m JKL m NML?
∠ = 67.5°, what is ∠
GJF ?
b. If NL = 14 inches and MN = 12 inches, what is ≅
JK? ANSWER:
A F, B G, C J,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
a.
67.5°
b. 12 in.
7.
Determine whether the triangles shown are
congruent. If so, name the corresponding parts
and write a congruence statement.
4.
STZ ?
≅
ANSWER:
∠S ∠Y,∠STZ ∠YTW,∠Z ∠W,
≅ ≅ ≅
ANSWER:
yes; A D, B E, C F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
8. KLM WXY
≅
Sample answer:
LKM ?
ABC DEF ≅
Δ ≅Δ
ANSWER:
K W, L X, M Y,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
5.
9. NPQ DEF
≅
EFD ?
≅
ANSWER:
ANSWER:
no
N D, P E, Q F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Name the corresponding parts in each pair of
congruent triangles. Then complete the
10. STAINED GLASS In the stained glass window
congruence statement.
shown, ABC ADC.
6. Δ ≅Δ
GJF ?
≅
ANSWER:
A F, B G, C J,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
a.
If BC = 26 centimeters, what is DC?
b.
If m ADC = 90°, and m DCA = 45°, what is
∠ ∠
7. m BAC?
∠
ANSWER:
a. 26 cm
b.
45°
11.
STZ ? 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 +
ANSWER: 1, what is the value of x?
∠S ∠Y,∠STZ ∠YTW,∠Z ∠W,
≅ ≅ ≅
ANSWER:
8.55 mm
8.
KLM WXY
≅ Determine whether the triangles shown are
LKM ?
≅ congruent. If so, name the corresponding parts
and write a congruence statement.
ANSWER:
12.
K W, L X, M Y,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
9. NPQ DEF
≅
EFD ?
≅
ANSWER:
N D, P E, Q F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
yes; P V, W Q, Z S,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
10. STAINED GLASS In the stained glass window
shown, ABC ADC. Sample answer:
Δ ≅Δ
ΔZWP ΔSQV
≅
13.
ANSWER:
a. If BC = 26 centimeters, what is DC? no
b.
If m ADC = 90°, and m DCA = 45°, what is
∠ ∠
m BAC? 14.
∠
ANSWER:
a. 26 cm
b.
45°
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 + ANSWER:
1, what is the value of x? no
ANSWER:
15.
8.55 mm
Determine whether the triangles shown are
congruent. If so, name the corresponding parts
and write a congruence statement.
12.
ANSWER:
yes; J P, L R, K Q,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
JLK PRQ
Δ ≅Δ
16.
If ABC DFG, name all corresponding sides.
Δ ≅Δ
ANSWER:
ANSWER:
yes; P V, W Q, Z S,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer: 17.
SOCIAL STUDIES The Khafre pyramid shown
below is composed of triangles.
ΔZWP ΔSQV
≅ (See page 636 of your text.)
If MNO NPO, m M m OPN=
Δ ≅Δ ∠ = 60°, and ∠
13. m O.
50°, find ∠
ANSWER:
70°
18.
GEOMETRY Hexagon ABCDEF has six
congruent sides. Segments , , and are
drawn on the hexagon, forming four triangles. Make
ANSWER: a conjecture about which triangles are congruent.
no Test your conjecture by measuring the sides and
angles of the triangles.
14.
ANSWER:
ANSWER: Triangles CBA and CDE are congruent. Triangles
11-2 Congruent Triangles
no CAF and CEF are congruent.
15. 19.
In the figure below, ΔFGI ΔLKM. Find the value
≅
of x. Round to the nearest tenth.
ANSWER:
ANSWER:
yes; J P, L R, K Q,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠ 3.6 cm
Sample answer:
20.
JLK PRQ CONSTRUCTION The roof contains two
Δ ≅Δ congruent triangles.
16.
If ABC DFG, name all corresponding sides.
Δ ≅Δ
ANSWER:
17.
SOCIAL STUDIES The Khafre pyramid shown a. Write a congruence statement involving the
below is composed of triangles. triangles. Then write a congruence statement
(See page 636 of your text.) involving a pair of corresponding sides.
If MNO NPO, m M m OPN=
Δ ≅Δ ∠ = 60°, and ∠ b. What is the distance from the right side of the
m O.
50°, find ∠
roof at the base to the center? Name the side.
ANSWER:
c. If the length of ,
is 4 feet longer than
70° what is the length of ?
18.
GEOMETRY Hexagon ABCDEF has six
ANSWER:
congruent sides. Segments , , and are
a. Δ
Sample answer: XYW ΔZYW;
drawn on the hexagon, forming four triangles. Make ≅ ≅
a conjecture about which triangles are congruent. b. 7 ft;
Test your conjecture by measuring the sides and c. 3 ft
angles of the triangles.
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.
ANSWER:
Triangles CBA and CDE are congruent. Triangles
CAF and CEF are congruent.
ANSWER:
19. In the figure below, FGI LKM. Find the value
Δ ≅Δ
of x. Round to the nearest tenth. Sample answer:
ANSWER:
eSolutions Manual - Powered by Cognero Page3
3.6 cm
20.
CONSTRUCTION The roof contains two
congruent triangles. GHI JKL, MNO PQR
≅ ≅
x for each pair of
ALGEBRA Find the value of
congruent triangles.
22.
a. Write a congruence statement involving the
triangles. Then write a congruence statement
involving a pair of corresponding sides.
b. What is the distance from the right side of the
roof at the base to the center? Name the side.
ANSWER:
c. If the length of ,
is 4 feet longer than 9
what is the length of ?
ANSWER:
23.
a. Δ
Sample answer: XYW ΔZYW;
≅ ≅
b. 7 ft;
c. 3 ft
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 ANSWER:
vertices, and write a congruence statement for each. 7
24.
OPEN ENDED Find a real-world example of
congruent triangles. Explain how you know the
triangles are congruent.
ANSWER:
See students’ work.
ANSWER:
Sample answer: 25. ERROR ANALYSIS Jade and Fernando are
writing a congruence statement for the congruent
triangles shown. Is either of them correct? Justify
your reasoning.
GHI JKL, MNO PQR
≅ ≅
x for each pair of
ALGEBRA Find the value of
congruent triangles.
22.
ANSWER:
No; corresponding vertices must be written in the
same order, so YXZ SRT, not STR and
∠ ≅∠ ∠
, not .
≅
ANSWER:
9
26. CHALLENGE Determine whether each statement
is true or false. If true, explain your reasoning. If
false, give a counterexample.
23.
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 ΔABC is 24 millimeters and the
perimeter of ΔRST is 24 millimeters. So, triangles
ABC and RST are congruent.
ANSWER:
7 ANSWER:
a. True; If the triangles are congruent, then the
24. OPEN ENDED Find a real-world example of corresponding sides have equal lengths. Therefore,
congruent triangles. Explain how you know the the sums of the lengths, or the perimeters, are equal.
triangles are congruent. b. True; If the triangles are congruent, then the
corresponding sides have equal lengths and the
ANSWER: corresponding bases and heights are equal.
See students’ work.
Therefore, the products • base • height, or the
25. ERROR ANALYSIS Jade and Fernando are
writing a congruence statement for the congruent areas, are equal.
triangles shown. Is either of them correct? Justify
c. False; Sample answer: the sides of ΔABC
your reasoning. measure 8 mm, 8mm, and 8 mm. The sides of ΔRST
measure 6 mm, 8 mm, and 10 mm.
27.
CHALLENGE Describe the scale factor of two
congruent triangles.
ANSWER:
The scale factor is 1:1 or 1.
28.
WRITING IN MATH Describe structures that
are built using congruent triangles. Explain why you
think congruent triangles are used.
ANSWER:
Sample answer: Congruent triangles are used in
bridges and roofs. Congruent triangles help balance
and support the structures.
ANSWER:
No; corresponding vertices must be written in the
same order, so YXZ SRT, not STR and 29. In the road sign, ∆ABC ∆DCB, AC is 2.5 meters
∠ ≅∠ ∠ long. What is the length of ?
, not .
≅
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.
A
b. Triangles WXY and HJK are congruent. So, their 1 meter
B
2.5 meters
areas are equal.
C
2.7 meters
c. The perimeter of ΔABC is 24 millimeters and the D
perimeter of ΔRST is 24 millimeters. So, triangles 2 meters
ABC and RST are congruent.
ANSWER:
ANSWER: B
a. True; If the triangles are congruent, then the
corresponding sides have equal lengths. Therefore, 30. Guy wires connected to a telephone pole create two
the sums of the lengths, or the perimeters, are equal. congruent triangles ∆PQR and ∆SQR. Find the
b. True; If the triangles are congruent, then the length of .
corresponding sides have equal lengths and the
corresponding bases and heights are equal.
Therefore, the products • base • height, or the
areas, are equal.
c. False; Sample answer: the sides of ΔABC
measure 8 mm, 8mm, and 8 mm. The sides of ΔRST
measure 6 mm, 8 mm, and 10 mm.
F
12 ft
27. CHALLENGE Describe the scale factor of two G
congruent triangles. 24 ft
H
48 ft
J
65 ft
ANSWER:
The scale factor is 1:1 or 1.
ANSWER:
F
28. WRITING IN MATH Describe structures that
are built using congruent triangles. Explain why you
think congruent triangles are used. 31. Which of the following statements is not true if
∆FGH ∆LMN?
ANSWER: A
Sample answer: Congruent triangles are used in
B
bridges and roofs. Congruent triangles help balance
and support the structures. C G M
∠ ∠
D
F N
∠ ∠
29. In the road sign, ∆ABC ∆DCB, AC is 2.5 meters
long. What is the length of ? ANSWER:
D
32. EXTENDED RESPONSE
Determine whether the
triangles shown are congruent. If so, name the
corresponding parts and write a congruence
statement.
A
1 meter
B
2.5 meters
C
2.7 meters
D
2 meters
ANSWER:
ANSWER:
B
yes; A D, B E, C F,
∠ ∠ ∠ ∠ ∠ ∠
30. Guy wires connected to a telephone pole create two , , ; Sample
congruent triangles ∆PQR and ∆SQR. Find the answer: ∆ABC ∆DEF
length of .
33.
Angles R and S are complementary. Find m R if
∠
m S= 65.7°.
∠
ANSWER:
24.3°
34. In a 45°-45°-90° triangle, a leg is 4 centimeters long.
Find the length of the hypotenuse to the nearest
tenth.
F
12 ft
G ANSWER:
24 ft 5.7 cm
H
48 ft
J
65 ft –4
35. BIRDS A mockingbird uses about 5 Joules of
ANSWER: energy to sing a song. Write the amount of energy
F the bird uses as an expression using a positive
exponent.
31. Which of the following statements is not true if
∆FGH ∆LMN? ANSWER:
A
B
C G M 36. CAR RENTAL The costs for renting a car from
∠ ∠ Able Car Rental and from Baker Car Rental are
D
F N
∠ ∠ shown in the table. For what mileage does Baker
ANSWER: have the better deal? Use the inequality 30 + 0.05x >
D 20 + 0.10x. Explain why this inequality works.
32. EXTENDED RESPONSE
Determine whether the
triangles shown are congruent. If so, name the
corresponding parts and write a congruence
statement.
ANSWER:
ANSWER: x < 200; Sample answer: The inequality finds at what
mileage Able’s charge is greater than Baker’s
yes; A D, B E, C F,
∠ ∠ ∠ ∠ ∠ ∠
charge.
, , ; Sample Graph each ordered pair on a coordinate plane.
answer: ∆ABC ∆DEF
37. F (4, 5)
33.
Angles R and S are complementary. Find m R if
∠
m S = 65.7°. ANSWER:
∠
ANSWER:
24.3°
34. In a 45°-45°-90° triangle, a leg is 4 centimeters long.
Find the length of the hypotenuse to the nearest
tenth.
ANSWER:
5.7 cm
–4
35. BIRDS A mockingbird uses about 5 Joules of
energy to sing a song. Write the amount of energy
38. G (2, 2)
the bird uses as an expression using a positive –
exponent.
ANSWER:
ANSWER:
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.
39.
C(–1, 4)
ANSWER:
ANSWER:
x < 200; Sample answer: The inequality finds at what
mileage Able’s charge is greater than Baker’s
charge.
Graph each ordered pair on a coordinate plane.
37.
F(4, 5)
40.
A (–3, –2)
ANSWER:
ANSWER:
38. G (2, 2)
–
ANSWER:
39.
C(–1, 4)
ANSWER:
40.
A (–3, –2)
ANSWER:
Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
1.
YZX
≅ ?
ANSWER:
∠L X, M Y, N Z,
≅∠ ∠ ≅∠ ∠ ≅∠
2.
RST GJK
≅
STR
≅ ?
ANSWER:
R G, S J, T K,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
3.
KITES In the umbrella kite shown, JLK
≅
NLM.
a. If m JKL m NML?
∠ = 67.5°, what is ∠
b. If NL = 14 inches and MN = 12 inches, what is
JK?
ANSWER:
a.
67.5°
b. 12 in.
Determine whether the triangles shown are
congruent. If so, name the corresponding parts
and write a congruence statement.
Name the corresponding parts in each pair of
congruent triangles. Then complete the 4.
congruence statement.
1.
ANSWER:
yes; A D, B E, C F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
YZX ?
≅ ABC DEF
Δ ≅Δ
ANSWER:
5.
∠L X, M Y, N Z,
≅∠ ∠ ≅∠ ∠ ≅∠
2.
RST GJK
≅
STR
≅ ?
ANSWER:
ANSWER: no
R G, S J, T K,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠ Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
3. KITES In the umbrella kite shown, JLK
≅
6.
NLM.
GJF ?
≅
ANSWER:
A F, B G, C J,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
a. If m JKL m NML?
∠ = 67.5°, what is ∠
b. If NL = 14 inches and MN = 12 inches, what is
JK? 7.
ANSWER:
a.
67.5°
b. 12 in.
Determine whether the triangles shown are
congruent. If so, name the corresponding parts STZ ?
≅
and write a congruence statement.
ANSWER:
4.
∠S ∠Y,∠STZ ∠YTW,∠Z ∠W,
≅ ≅ ≅
8.
KLM WXY
≅
LKM ?
≅
ANSWER:
ANSWER:
K W, L X, M Y,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
yes; A D, B E, C F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
ABC DEF
Δ ≅Δ
9. NPQ DEF
≅
EFD ?
≅
5.
ANSWER:
N D, P E, Q F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
10. STAINED GLASS In the stained glass window
ANSWER: shown, ABC ADC.
no Δ ≅Δ
Name the corresponding parts in each pair of
congruent triangles. Then complete the
congruence statement.
6.
a. If BC = 26 centimeters, what is DC?
GJF ?
≅
b. If m ADC m DCA
∠ = 90°, and ∠ = 45°, what is
ANSWER: m BAC?
∠
A F, B G, C J,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER:
a. 26 cm
b.
45°
7.
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?
ANSWER:
STZ ? 8.55 mm
≅
ANSWER: Determine whether the triangles shown are
congruent. If so, name the corresponding parts
∠S ∠Y,∠STZ ∠YTW,∠Z ∠W,
≅ ≅ ≅
and write a congruence statement.
12.
8.
KLM WXY
≅
LKM ?
≅
ANSWER:
K W, L X, M Y,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
9.
NPQ DEF ANSWER:
≅
EFD ?
≅
yes; P V, W Q, Z S,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
ANSWER: Sample answer:
ΔZWP ΔSQV
≅
N D, P E, Q F,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
13.
10. STAINED GLASS In the stained glass window
shown, ABC ADC.
Δ ≅Δ
ANSWER:
no
14.
a. If BC = 26 centimeters, what is DC?
b.
If m ADC = 90°, and m DCA = 45°, what is
∠ ∠
m BAC?
∠
ANSWER:
a. 26 cm ANSWER:
b. no
45°
15.
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?
ANSWER:
8.55 mm
ANSWER:
Determine whether the triangles shown are
yes; J P, L R, K Q,
congruent. If so, name the corresponding parts ∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
and write a congruence statement. JLK PRQ
Δ ≅Δ
12.
16.
If ABC DFG, name all corresponding sides.
Δ ≅Δ
ANSWER:
17.
SOCIAL STUDIES The Khafre pyramid shown
below is composed of triangles.
(See page 636 of your text.)
ANSWER:
If MNO NPO, m M m OPN=
Δ ≅Δ ∠ = 60°, and ∠
yes; P V, W Q, Z S,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠ m O.
50°, find ∠
Sample answer:
ANSWER:
ΔZWP ΔSQV
≅
70°
13.
18. GEOMETRY Hexagon ABCDEF has six
congruent sides. Segments , , and 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.
ANSWER:
no
14.
ANSWER:
Triangles CBA and CDE are congruent. Triangles
CAF and CEF are congruent.
19.
In the figure below, ΔFGI ΔLKM. Find the value
≅
of x. Round to the nearest tenth.
ANSWER:
no
15.
ANSWER:
3.6 cm
20.
CONSTRUCTION The roof contains two
congruent triangles.
ANSWER:
yes; J P, L R, K Q,
∠ ≅∠ ∠ ≅∠ ∠ ≅∠
Sample answer:
JLK PRQ
Δ ≅Δ
16.
If ABC DFG, name all corresponding sides.
Δ ≅Δ a. Write a congruence statement involving the
ANSWER: triangles. Then write a congruence statement
involving a pair of corresponding sides.
b. What is the distance from the right side of the
roof at the base to the center? Name the side.
17. SOCIAL STUDIES The Khafre pyramid shown
below is composed of triangles. c. If the length of ,
is 4 feet longer than
(See page 636 of your text.) what is the length of ?
If MNO NPO, m M m OPN=
Δ ≅Δ ∠ = 60°, and ∠
m O. ANSWER:
50°, find ∠
a. Δ
Sample answer: XYW ΔZYW;
≅ ≅
ANSWER:
b. 7 ft;
70°
c. 3 ft
18. GEOMETRY Hexagon ABCDEF has six
21.
congruent sides. Segments , , and BUTTERFLIES Butterfly wings are triangular in
are shape. Using the art of the butterfly as a model,
drawn on the hexagon, forming four triangles. Make draw two sets of congruent triangles, label the
a conjecture about which triangles are congruent. vertices, and write a congruence statement for each.
Test your conjecture by measuring the sides and
angles of the triangles.
ANSWER:
ANSWER: Sample answer:
Triangles CBA and CDE are congruent. Triangles
CAF and CEF are congruent.
19.
In the figure below, ΔFGI ΔLKM. Find the value
≅
of x. Round to the nearest tenth.
GHI JKL, MNO PQR
≅ ≅
ANSWER: x for each pair of
3.6 cm ALGEBRA Find the value of
congruent triangles.
22.
20. CONSTRUCTION The roof contains two
congruent triangles.
a. Write a congruence statement involving the ANSWER:
triangles. Then write a congruence statement 9
involving a pair of corresponding sides.
b. What is the distance from the right side of the
23.
roof at the base to the center? Name the side.
c. If the length of ,
is 4 feet longer than
what is the length of ?
ANSWER:
a. Δ
Sample answer: XYW ΔZYW;
≅ ≅
b. 7 ft; ANSWER:
11-2 Congruent Triangles
c. 3 ft 7
21. 24.
BUTTERFLIES Butterfly wings are triangular in OPEN ENDED Find a real-world example of
shape. Using the art of the butterfly as a model, congruent triangles. Explain how you know the
draw two sets of congruent triangles, label the triangles are congruent.
vertices, and write a congruence statement for each.
ANSWER:
See students’ work.
25.
ERROR ANALYSIS Jade and Fernando are
writing a congruence statement for the congruent
triangles shown. Is either of them correct? Justify
your reasoning.
ANSWER:
Sample answer:
GHI JKL, MNO PQR
≅ ≅
x for each pair of
ALGEBRA Find the value of
ANSWER:
congruent triangles.
22. No; corresponding vertices must be written in the
same order, so YXZ SRT, not STR and
∠ ≅∠ ∠
, not .
≅
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
ANSWER: perimeters are equal.
9
b. Triangles WXY and HJK are congruent. So, their
areas are equal.
23.
c. The perimeter of ΔABC is 24 millimeters and the
perimeter of ΔRST is 24 millimeters. So, triangles
ABC and RST are congruent.
ANSWER:
a. True; If the triangles are congruent, then the
corresponding sides have equal lengths. Therefore,
the sums of the lengths, or the perimeters, are equal.
ANSWER: b. True; If the triangles are congruent, then the
7
corresponding sides have equal lengths and the
corresponding bases and heights are equal.
24. OPEN ENDED Find a real-world example of
congruent triangles. Explain how you know the Therefore, the products • base • height, or the
triangles are congruent.
eSolutions Manual - Powered by Cognero areas, are equal. Page4
ANSWER:
c. False; Sample answer: the sides of ΔABC
See students’ work. measure 8 mm, 8mm, and 8 mm. The sides of ΔRST
measure 6 mm, 8 mm, and 10 mm.
25. ERROR ANALYSIS Jade and Fernando are
27.
writing a congruence statement for the congruent CHALLENGE Describe the scale factor of two
triangles shown. Is either of them correct? Justify congruent triangles.
your reasoning.
ANSWER:
The scale factor is 1:1 or 1.
28.
WRITING IN MATH Describe structures that
are built using congruent triangles. Explain why you
think congruent triangles are used.
ANSWER:
Sample answer: Congruent triangles are used in
bridges and roofs. Congruent triangles help balance
and support the structures.
29. In the road sign, ∆ABC ∆DCB, AC is 2.5 meters
long. What is the length of ?
ANSWER:
No; corresponding vertices must be written in the
same order, so YXZ SRT, not STR and
∠ ≅∠ ∠
, not .
≅
A
26. 1 meter
CHALLENGE Determine whether each statement
B
is true or false. If true, explain your reasoning. If 2.5 meters
C
false, give a counterexample. 2.7 meters
D
a. Triangles EFG and KLM are congruent. So, their 2 meters
perimeters are equal.
b. Triangles WXY and HJK are congruent. So, their ANSWER:
B
areas are equal.
c. The perimeter of ΔABC is 24 millimeters and the
perimeter of ΔRST is 24 millimeters. So, triangles 30. Guy wires connected to a telephone pole create two
ABC and RST are congruent. congruent triangles ∆PQR and ∆SQR. Find the
length of .
ANSWER:
a. True; If the triangles are congruent, then the
corresponding sides have equal lengths. Therefore,
the sums of the lengths, or the perimeters, are equal.
b. True; If the triangles are congruent, then the
corresponding sides have equal lengths and the
corresponding bases and heights are equal.
Therefore, the products • base • height, or the
F
12 ft
areas, are equal. G
24 ft
c. False; Sample answer: the sides of ΔABC H
measure 8 mm, 8mm, and 8 mm. The sides of ΔRST 48 ft
J
measure 6 mm, 8 mm, and 10 mm. 65 ft
ANSWER:
27.
CHALLENGE Describe the scale factor of two F
congruent triangles.
31. Which of the following statements is not true if
ANSWER: ∆FGH ∆LMN?
The scale factor is 1:1 or 1. A
28.
WRITING IN MATH Describe structures that B
are built using congruent triangles. Explain why you
C G M
think congruent triangles are used. ∠ ∠
D
F N
∠ ∠
ANSWER:
Sample answer: Congruent triangles are used in ANSWER:
bridges and roofs. Congruent triangles help balance D
and support the structures.
32. EXTENDED RESPONSE
Determine whether the
triangles shown are congruent. If so, name the
29. In the road sign, ∆ABC ∆DCB, AC is 2.5 meters corresponding parts and write a congruence
long. What is the length of ? statement.
A ANSWER:
1 meter
B
2.5 meters
yes; A D, B E, C F,
C ∠ ∠ ∠ ∠ ∠ ∠
2.7 meters
D , , ; Sample
2 meters
answer: ∆ABC ∆DEF
ANSWER:
B
33.
Angles R and S are complementary. Find m R if
∠
m S= 65.7°.
∠
30. Guy wires connected to a telephone pole create two
congruent triangles ∆PQR and ∆SQR. Find the ANSWER:
length of . 24.3°
34. In a 45°-45°-90° triangle, a leg is 4 centimeters long.
Find the length of the hypotenuse to the nearest
tenth.
ANSWER:
5.7 cm
–4
35. BIRDS A mockingbird uses about 5 Joules of
F energy to sing a song. Write the amount of energy
12 ft the bird uses as an expression using a positive
G
24 ft exponent.
H
48 ft
J
65 ft
ANSWER:
ANSWER:
F
31.
Which of the following statements is not true if 36. CAR RENTAL The costs for renting a car from
∆FGH ∆LMN? Able Car Rental and from Baker Car Rental are
A shown in the table. For what mileage does Baker
have the better deal? Use the inequality 30 + 0.05x >
B 20 + 0.10x. Explain why this inequality works.
C G M
∠ ∠
D
F N
∠ ∠
ANSWER:
D
32. EXTENDED RESPONSE
Determine whether the
triangles shown are congruent. If so, name the
corresponding parts and write a congruence
statement.
ANSWER:
x < 200; Sample answer: The inequality finds at what
mileage Able’s charge is greater than Baker’s
charge.
Graph each ordered pair on a coordinate plane.
37.
ANSWER: F(4, 5)
ANSWER:
yes; A D, B E, C F,
∠ ∠ ∠ ∠ ∠ ∠
, , ; Sample
answer: ∆ABC ∆DEF
33.
Angles R and S are complementary. Find m R if
∠
m S= 65.7°.
∠
ANSWER:
24.3°
34. In a 45°-45°-90° triangle, a leg is 4 centimeters long.
Find the length of the hypotenuse to the nearest
tenth.
38. G (2, 2)
–
ANSWER:
5.7 cm
ANSWER:
–4
35. BIRDS A mockingbird uses about 5 Joules of
energy to sing a song. Write the amount of energy
the bird uses as an expression using a positive
exponent.
ANSWER:
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
39. C ( 1, 4)
have the better deal? Use the inequality 30 + 0.05x > –
20 + 0.10x. Explain why this inequality works.
ANSWER:
ANSWER:
x < 200; Sample answer: The inequality finds at what
mileage Able’s charge is greater than Baker’s
40.
A (–3, –2)
charge.
Graph each ordered pair on a coordinate plane. ANSWER:
37.
F(4, 5)
ANSWER:
38. G (2, 2)
–
ANSWER:
39.
C(–1, 4)
ANSWER:
40.
A (–3, –2)
ANSWER:
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