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LESSON
Class
Reteach
7-5 Triangles
You can classify triangles by the measures of their angles.
An acute triangle has only acute angles.
A right triangle has one right angle.
An obtuse triangle has one obtuse angle.
To classify the triangle below by its angles, you first need to find the
measure of all the angles.
C
46°
The sum of the angles of a triangle is 180°.
34°
A
B
To find the unknown angle, first find the sum of the two known angles.
34 ! 46 " 80
Then subtract the sum from 180.
180 # 80 " 100
The difference is the measure of the third angle.
The measures of the angles are 34°, 46°, and 100°. So the triangle
is obtuse.
Find each unknown angle. Then classify each triangle.
1.
2. A
Y
40°
75°
77°
X
50°
Z
B
3.
4.
90°
30°
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C
60°
60°
50
Holt Middle School Math Course 1
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LESSON
Class
Reteach
7-5 Triangles (continued)
You can also classify triangles by the lengths of their sides.
A scalene triangle has no congruent sides.
An isosceles triangle has at least 2 congruent sides.
An equilateral triangle has 3 congruent sides.
The sum of the lengths of the sides of the triangle below is 18.6 cm.
To classify the triangle below by its sides, you first need to find the
length of the unknown side.
A
7.3 cm
3.5 cm
B
C
?
To find the unknown length, first find the sum of the two known
lengths.
7.3 ! 3.5 " 10.8
Then subtract this sum from the sum of all three sides.
18.6 # 10.8 " 7.8
The length of the third side is 7.8 cm.
The lengths of the sides of the triangle are 3.5 cm, 7.3 cm, and
7.8 cm, so the triangle is scalene.
Find each missing length. Then classify each triangle.
3.
4.
L
6.7 in.
X
6.7 in.
9 in.
J
?
?
K
Y
4 in. Z
The sum of the lengths of the
sides is 22 inches.
The sum of the lengths of the
sides is 20.1 inches.
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51
Holt Middle School Math Course 1
Practice B
7-5 Triangles
Practice A
7-5 Triangles
LESSON
LESSON
Use the diagram to find the measure of each indicated angle.
Classify each triangle as acute, obtuse, or right.
1.
2.
3.
60°
70°
90°
30°
30°
60°
right
50°
acute
5.
2 in.
2 in.
3 in.
6.
5 cm
5 cm
90°
4. !EBA
90°
5. !ACB
45°
B
10 in.
7. The sum of the lengths
of the three sides is
15 cm.
10 in.
?
70°
45°
8 ft
6 ft
3 cm
equilateral
35°
?
45°
8. The sum of the lengths
of the three sides is
22 ft.
5 cm
9.
50°
90°
C
A
6. The sum of the lengths
of the three sides is
30 in.
scalene
8.
?
45°
45°
Find the measure of the unknown angle.
7.
D
E
Classify each triangle using the given information.
5 ft
equilateral
45°
3. !ABC
3 ft
isosceles
2. !DAC
120° 30°
4 ft
5 cm
90°
obtuse
Classify each triangle as scalene, isosceles, or equilateral.
4.
1. !CBD
60°
scalene
isosceles
9. The angles of a triangular sail measure 90°, 30°, and 60°. Its
sides measure approximately 2 feet, 3.5 feet, and 4 feet. Classify
the triangular shape of the sail in two different ways.
35°
110°
It is a right triangle; It is a scalene triangle.
10. One side of an equilateral triangle measures 4 cm. What are the
lengths of the other two sides of the triangle?
10. Two angles in one triangle are congruent to two angles in
another triangle. What can you conclude about the third angle
in both triangles?
They both measure 4 cm.
They are congruent.
11. Karen says the angles of her triangle measure 90°, 50°, and
60°. Explain why this is impossible.
The sum of those angles is 200°, and the sum of the angles in any
triangle is always 180°.
47
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Holt Middle School Math Course 1
LESSON
Use the diagram to find the measure of each indicated angle.
1. !DEC
79°
B
2. !ACB
52°
70°
3. !BCE
117°
52°
You can classify triangles by the measures of their angles.
An acute triangle has only acute angles.
A right triangle has one right angle.
An obtuse triangle has one obtuse angle.
E
11°
58°
To classify the triangle below by its angles, you first need to find the
measure of all the angles.
11°
A
C
C
D
46°
The lengths of sides !
AB
! and B
!C
! are given for "ABC. Use the
sum of the lengths of the three sides to calculate the length of
side !
AC
! and classify each triangle.
5. AB ! 1.5; BC ! 4.8;
sum ! 10
1
1
6. AB ! 2!4!; BC ! 2!4!;
3
sum ! 6!4!
7. AB ! 638; BC ! 181;
AC ! 181;
scalene
equilateral
isosceles
1
!B ! !2! • !A
To find the unknown angle, first find the sum of the two known angles.
Then subtract the sum from 180.
180 # 80 ! 100
The difference is the measure of the third angle.
The measures of the angles are 34°, 46°, and 100°. So the triangle
is obtuse.
The measures of !A and !B are given for "ABC. Calculate the
measure of !C and classify each triangle.
9. !A ! 60°;
B
34 " 46 ! 80
sum ! 1,000
AC ! 3.7;
!B ! 136.5°
The sum of the angles of a triangle is 180°.
34°
A
1
AC ! 2!4!;
8. !A ! 22.17°;
Holt Middle School Math Course 1
Reteach
7-5 Triangles
Practice C
7-5 Triangles
LESSON
4. !EBC
48
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10. !A ! 26.5°;
Find each unknown angle. Then classify each triangle.
!B ! 3 • !A
!C ! 21.33°;
!C ! 90°;
!C ! 74°;
obtuse
right
acute
1.
2. A
Y
40°
75°
77°
X
11. Starting from the airport, a pilot flew directly north for 112 miles.
Then she made a 90° turn and flew east for 84 miles to reach
Bear Lake. If she flew in a straight line, in which direction did the
pilot fly to get back to the airport? If she flew a total of 336 miles,
how far is the airport from Bear Lake?
C
50°
Z
B
28°, acute
90°, right
3.
4.
60°
She flew southwest to the airport; 140 miles
90°
12. !A of "ABC is congruent to !G of "FGH. !B of "ABC and
!F of "FGH are supplementary. !C measures 29°, and !A
measures 37°. What is the measure of each angle in "FGH?
49
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60°
60°, right
!G ! 37°; !F ! 66°; !H ! 77°
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30°
Holt Middle School Math Course 1
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150
60°, acute
50
Holt Middle School Math Course 1
Holt Middle School Math Course 1
Challenge
7-5 Square Legs
Reteach
7-5 Triangles (continued)
LESSON
LESSON
You can also classify triangles by the lengths of their sides.
About 2,400 years ago a Greek philosopher
and mathematician named Pythagoras proved
a very important rule for triangles. Today
people all over the world study and use this
rule named in his honor—the Pythagorean
Theorem. It states the relationship between
the side lengths of a right triangle.
A scalene triangle has no congruent sides.
An isosceles triangle has at least 2 congruent sides.
An equilateral triangle has 3 congruent sides.
The sum of the lengths of the sides of the triangle below is 18.6 cm.
To classify the triangle below by its sides, you first need to find the
length of the unknown side.
A
7.3 cm
3.5 cm
B
Pythagoras
C
?
PYTHAGOREAN THEOREM
To find the unknown length, first find the sum of the two known
lengths.
In a right triangle, the square of the length
of the hypotenuse is equal to the sum of
the squares of the lengths of the legs.
7.3 " 3.5 ! 10.8
The legs are the
sides that form the
right angle
Then subtract this sum from the sum of all three sides.
18.6 # 10.8 ! 7.8
2
2
(leg) " (leg) ! (hypotenuse)
The hypoteneuse is
the side opposite
the right angle
2
The length of the third side is 7.8 cm.
a2 " b2 !
The lengths of the sides of the triangle are 3.5 cm, 7.3 cm, and
7.8 cm, so the triangle is scalene.
2
2
3
"
4
9
" 16
c2
c=5
a=3
2
!
5
!
25
b=4
Find each missing length. Then classify each triangle.
3.
4.
L
6.7 in.
X
9 in.
J
Use the Pythagorean Theorem to find the length of the missing
side for each right triangle below.
6.7 in.
?
?
K
4 in. Z
The sum of the lengths of the
sides is 22 inches.
The sum of the lengths of the
sides is 20.1 inches.
6.7 inches, equilateral
c = 13
a=?
Y
c=?
a=7
b = 24
b = 12
9 inches, isosceles
a!5
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51
Holt Middle School Math Course 1
Problem Solving
7-5 Triangles
c ! 25
52
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Holt Middle School Math Course 1
Puzzles, Twisters & Teasers
7-5 Find the Measure
LESSON
LESSON
Use the triangle diagram to answer each question.
1. Classify triangle ABC. What is the
measure of the missing angle?
M
3 cm
"ABC is an acute triangle, 70°
2. Classify triangle XYZ. What is the
measure of the missing angle?
2 in.
3 cm
O
G
N
"XYZ is an obtuse triangle, 100°
Find the missing angle measure in each triangle. When you
have completed the problem, look at the blank lines at the
bottom of the page. Notice if any of the angle degrees of the
letters match the degrees under the blank lines. If they do, write
the letter of the angle on the line. The resulting message will be
the answer to the riddle.
E
4 in.
3 in.
F
X
1.
2.
x°
29°
3. If triangle MNO is an equilateral
triangle, what is the measure of the
missing side?
C
!N
O
! ! 3 cm
50°
60°
B
51°
Z
70°
Y
K
5. Classify triangle EFG.
4. What is the complement of !XYZ?
"EFG is a scalene triangle
39°
70°
x°
J
x ! 40°
x°
T
V
R
5.
Y
x°
87°
30°
C
measures of the angles in any
35°
x ! 105°
x ! 45°
F
120°
Because the sum of the
40°
x°
4.
7. What is the supplement of !ABC?
6. Explain how you can find the
measure of !CAB.
3.
V
A
x°
33°
W
x ! 60°
M
N
x ! 120°
triangle is 180°, subtract the
measure of !ACB and !ABC
from 180°.
6.
7.
S
Circle the letter of the correct answer.
9. Which of the following is not true of
all right triangles?
F The sum of the measures of the
angles is 180°.
G Two of its angles are
!
complementary angles.
H At least two of its angles are
acute.
J The side with the greatest length
is opposite the right angle.
8. Which of the following statements is
always true?
A A right triangle is a scalene
triangle.
B An equilateral triangle is an
!
isosceles triangle.
C An isosceles triangle is an obtuse
triangle.
D A right triangle is an acute
triangle.
60°
30°
Z
x!
53
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Holt Middle School Math Course 1
x°
U
A
50°
I
35°
115°
z ! 55°
L
x ! 68°
y ! 62°
x ° 112°
L
Why do you like studying about volcanoes?
I
50°
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All rights reserved.
y°
z°
T ’
105°
S
A
L
A
V
A
F
U
N
55°
62°
68°
62°
45°
62°
60°
115°
120°
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All rights reserved.
151
54
Holt Middle School Math Course 1
Holt Middle School Math Course 1