Download Triangles

7.5
Triangles
7.5
OBJECTIVES
1.
2.
3.
4.
5.
Find the measure of the third angle of a triangle
Classify triangles as acute, obtuse, or right
Classify triangles as equilateral, isosceles, or scalene
Recognize similar triangles
Apply similar triangles to find an unknown length.
Now that you know something about angles, it is interesting to again look at triangles. Why
is this shape called a triangle?
Literally, triangle means “three angles.”
The same classifications we used for angles can be used for triangles. If a triangle has a
right angle, we call it a right triangle.
If it has three acute angles, it is called an acute triangle.
If it has an obtuse angle, it is called an obtuse triangle.
Example 1
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Identifying an Acute Triangle
Which of the following triangles is acute?
E
B
A
C
D
X
F
Z
Y
Only DEF is an acute triangle. Both ABC and XYZ have one obtuse angle.
585
586
CHAPTER 7
GEOMETRY AND MEASURE
CHECK YOURSELF 1
Which of the following triangles is obtuse?
Z
Y
B
E
C
A
F
D
X
We can also classify triangles based on how many angles have the same measure.
A triangle is called an equilateral triangle if all three angles have the same measure.
A triangle is called an isosceles triangle if two angles have the same measure.
A triangle is called a scalene triangle if no two angles have the same measure.
Example 2
Labeling Types of Triangles
B
E
40
60
Z
60
70
A
NOTE Notice that each of
these triangles can be classified
in different ways. XYZ is a right
triangle, but it is also scalene.
60
70
C
D
60
30
F
Y
X
ABC is an isosceles triangle because two of the angles have the same measure. And
DEF is an equilateral triangle because all three angles have the same measure. XYZ is
a scalene triangle because no two angles have the same measure.
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Of the following triangles, which are equilateral? Isosceles? Scalene?
TRIANGLES
SECTION 7.5
587
CHECK YOURSELF 2
Label each triangle as equilateral, isosceles, or scalene.
60
70
80
30
100
60
(1)
60
40
(2)
40
(3)
Go back and look at the sum of the angles inside each of the triangles in the last example. You will note that they always add up to 180°. No matter how we draw a triangle, the
sum of the three angles inside the triangle will always be 180°.
Here is an experiment that might convince you that this is always the case.
1. Using a straight edge, draw any triangle you wish on a sheet of paper.
2. Use scissors to cut out the triangle.
3. Rip the three vertices off of the triangle.
4. Lay the three vertices (with the points of the triangle touching) together. They will
always form a straight angle, which we saw in the previous section has a measure of
180°.
Rules and Properties:
Angles of a Triangle
For any triangle ABC,
mA mB mC 180°
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Example 3
Finding an Angle Measure
Find the measure of the third angle in this triangle.
?
53
68
588
CHAPTER 7
GEOMETRY AND MEASURE
We need the three measurements to add to 180°, so we add the two given measurements
(53° 68° 121°). Then we subtract that from 180° (180° 121° 59°). This gives us
the measure of the third angle, 59°.
CHECK YOURSELF 3
Find the measure of ABC.
B
70
35
C
A
Definitions: Similar Triangles
If the measurements of the three angles in two different triangles are the same,
we say the two triangles are similar triangles.
Example 4
Identifying Similar Triangles
Which two triangles are similar?
B
E
40
40
Z
70
Y
X
Although they are of different size, ABC and XYZ are similar because they have the
same angle measurements.
CHECK YOURSELF 4
Find the two triangles that are similar.
E
Z
B
45
30
60
A
C
D
F
Y
X
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ABC XYZ
40
F
C
A
NOTE This can be written
50
D
70 70
70
TRIANGLES
SECTION 7.5
589
Finally, let’s return to an idea that we first saw in Chapter 5.
Rules and Properties:
Similar Triangles
If two triangles are similar, their corresponding sides have the same ratio.
The most common use of this property of similar triangles occurs when you wish to find the
height of a tall object. Example 5 illustrates this application of similar triangles.
Example 5
Finding the Height of a Tree
If a man who is 180 cm tall casts a shadow that is 60 cm long, how tall is a tree that casts a
shadow that is 9 m long?
Note that, because of the sun, the man and his shadow forms a similar triangle to the tree
and its shadow. Because of this, we can use the common ratio to find the height of the tree.
xm
180 cm
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60 cm
9m
180 cm
xm
60 cm
9m
x
(180 ⋅ 9)
60 m
x 27 m
The tree is 27 m tall.
CHAPTER 7
GEOMETRY AND MEASURE
CHECK YOURSELF 5
If a man who is 160 cm tall casts a shadow that is 120 cm long, how tall is a building
that casts a shadow that is 60 m long?
CHECK YOURSELF ANSWERS
1. XYZ
2. (1) Scalene; (2) equilateral; (3) isosceles
4. DEF and XZY
5. 80 m
3. 75°
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590
Name
Exercises
7.5
Section
Date
In exercises 1 to 4, label the triangles as acute or obtuse.
1.
ANSWERS
2.
1.
2.
3.
4.
3.
4.
5.
6.
7.
8.
In exercises 5 to 10, label the triangles as equilateral, isosceles, or scalene.
5.
9.
10.
6.
53
60
60
7.
8.
60
40
60
70
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9.
10.
25
40
120
130
591
ANSWERS
11.
In exercises 11 to 16, find the missing angle and then label the triangle as equilateral,
isosceles, or scalene.
12.
11.
13.
12.
14.
15.
30
50
120
16.
17.
18.
13.
14.
19.
20.
45
30
15.
16.
67
46
65
50
For each triangle shown, find the indicated angle.
17. Find mC
18. Find mC
B
C
82
71
A
C
19. Find mA
23
B
20. Find mB
B
B
39
A
18
A
592
C
31
15
C
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61
A
ANSWERS
21.
21. Find mB
22. Find mA
B
22.
C
23.
A
24.
18
B
63
A
25.
26.
C
27.
In exercises 23 and 24, assume that the given triangle is isosceles.
23. Find mA and mC
24. Find mD and mF
E
B
110
8
72
8
15
A
15
C
F
D
25. Which two triangles are similar?
60
60
80
50
70
50
(b)
(a)
(c)
26. Which two triangles are similar?
30
(a)
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25
60
(c)
(b)
In exercises 27 to 32, the two triangles shown are similar. Find the indicated side.
27. Find v
S
V
3
12
R
5
T
U
v
W
593
ANSWERS
28.
28. Find f
29.
E
B
30.
4
6
31.
20
f
C
A
F
D
29. Find g
K
H
g
24
I
L
14
G
28
J
30. Find m
Q
N
m
10
O
M
35
20
R
P
T
W
38.7
50.4
V
S
594
t
U
30.1
X
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31. Find t
ANSWERS
32.
32. Find e
33.
D
G
e
E
C
34.
28.5
45.6
40.5
H
F
35.
36.
In exercises 33 to 36, first show that the two triangles are similar. Then find the indicated
side. Round your answer to the nearest hundredth.
33. Find KL
37.
34. Find PQ
P
K
?
?
59
L
70
N
M
78
65
Q
S
R
92
O
98
T
35. Find VX
36. Find AC
V
A
48
?
40
?
Y
W
D
X
B
C
60
82
51
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Z
36
E
Find the indicated side. If necessary, round to the nearest tenth of a unit.
___
37. Find DE
D
?
B
4
A
8
C
12
E
595
ANSWERS
__
38.
38. Find IJ
39.
I
40.
?
G
41.
2
42.
F
H
6
J
9
___
39. Find KL
L
M
?
32
K
N
38
O
52
___
40. Find PQ
Q
?
R
14
P
S
45
T
32
41. Given:
mBCA mDEA
___
Find DE
D
B
?
17
C
31
E
14
42. Given:
__mGHF mIJF
Find IJ
I
?
G
27
F
596
41
H
46
J
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A
ANSWERS
43. A light pole casts a shadow that measures 4 ft. At the same time, a yardstick casts a
shadow that is 9 in. long. How tall is the pole?
43.
44.
45.
46.
47.
44. A tree casts a shadow that measures 5 m. At the same time, a meter stick casts a
shadow that is 0.4 m long. How tall is the tree?
45. Use the ideas of similar triangles to determine the height of a pole or tree on your
campus. Work with one or two partners.
In exercises 46 to 48, one side of the triangle has been extended, forming what is called an
“exterior angle.” In each case, find the measure of the indicated exterior angle.
46.
85
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?
58
47.
44
?
83
597
ANSWERS
48.
48.
61
?
98
49.
49. What do you observe from exercises 46 to 48? Write a general conjecture about an
50.
exterior angle of a triangle.
51.
50. Write an argument to show that an equilateral triangle cannot have a right
angle.
52.
51. Argue that, given an equilateral triangle, the measure of each angle must
53.
be 60°.
54.
52. Argue that a triangle cannot have more than one obtuse angle.
53. Is it possible to have an isosceles right triangle? If such a triangle exists, what can be
said about the angles? Defend your statements.
If the sum of the measures of two angles is 90°, the two angles are said to be
complementary. If A and B are complementary angles, A is said to be the
complement of B, and B is said to be the complement of A.
54. Create an argument to support the following statement:
If ABC is a right triangle, with mC 90°, then A and B must be acute and
complementary.
1. Acute
3. Acute
5. Equilateral
7. Isosceles
9. Isosceles
11. 30°; isosceles
13. 45°; isosceles
15. 67°; isosceles
17. 37°
19. 123°
21. 27°
23. A 35°; C 35°
25. b and c
27. 20
29. 12
31. 39.2
33. 77.12
35. 65.60
37. 10
39. 55.4
41. 24.7
43. 16 ft
45.
47. 127°
49.
51.
598
53.
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Answers