Download Triangles

Page 1 of 6
Triangles
BEFORE
Now
WHY?
You classified angles.
You’ll classify triangles.
So you can classify Research
Triangle Park, as in Ex. 26.
You can find the sum of the angle measures in a triangle.
Word Watch
acute triangle, p. 485
right triangle, p. 485
obtuse triangle, p. 485
congruent sides, p. 486
equilateral triangle, p. 486
isosceles triangle, p. 486
scalene triangle, p. 486
1 Cut a triangle from the corner of a
piece of paper. Label the corners
aA, aB, and aC.
2 Tear the corners off the triangle.
3 Rearrange aA, aB, and aC so that
they are adjacent. Then make a conclusion
about the angles of the triangle.
In the activity, you found that the sum of the angle measures in TABC
is 180. The sum of the angle measures in any triangle is 180.
with
EXAMPLE
Reading
Triangles are named by their
vertices. The vertices of the
triangle in the activity are A,
B, and C, so the triangle can
be named with the notation
TABC. This notation is read
“triangle ABC.”
1
Finding an Angle Measure in a Triangle
Find the value of x in the triangle shown.
Sum of angle measures
in a triangle is 180.
x 83 26 180
x 109 180
26
Add 83 and 26.
x 71
Subtract 109 from each side.
83
x
ANSWER The value of x is 71.
Interior and Exterior Angles The three angles of any triangle are called
interior angles. The sides of a triangle can be extended to form angles
outside of the triangle that are adjacent and supplementary to the
interior angles. These angles are called exterior angles. You can use the
measures of interior angles to find the measures of exterior angles.
exterior angle
interior angles
484
Chapter 10
Geometric Figures
Page 2 of 6
2
EXAMPLE
Finding the Measure of an Exterior Angle
Find the value of y in the figure.
55
To find the value of y, use the fact that
adjacent interior and exterior angles
of a triangle are supplementary.
y 35 180
y 145
y
35
Definition of supplementary angles
Subtract 35 from each side.
ANSWER The value of y is 145.
Your turn now
1.
Find the value of x or y.
43
2.
x
108
38
79
34
y
You can classify any triangle by the measures of its interior angles.
Classifying Triangles by Angle Measures
Acute Triangle
Right Triangle
Obtuse Triangle
An acute triangle
has three acute angles.
A right triangle has
one right angle.
An obtuse triangle
has one obtuse angle.
60
50
120
35
70
EXAMPLE
3
50
25
40
Classifying a Triangle by Angle Measures
Classify the triangle formed by the plant
hanger by its angle measures.
The triangle has one right angle, so it is
a right triangle.
Lesson 10.3
Triangles
485
Page 3 of 6
with
Reading
The notation AB
& is read
“line segment AB” and
represents a line segment
with endpoints A and B.
The notation AB represents
the length of AB
&.
Congruent Sides Just as congruent angles
have the same measure, congruent sides of
a triangle have the same length. You can use
special marks to indicate that two sides or two
angles of a triangle are congruent. In the triangle
&* c XZ
&* and
at the right, the marks show that XY
that aY c aZ. You can classify any triangle
by the lengths of its sides.
X
Y
Z
Classifying Triangles by Side Lengths
Equilateral Triangle
Isosceles Triangle
Scalene Triangle
An equilateral triangle
has 3 congruent sides.
An isosceles triangle
has at least 2 congruent
sides.
A scalene triangle
has no congruent
sides.
8m
9 in.
5 ft
9 in.
5m
4
EXAMPLE
11 ft
8m
9 in.
14 ft
Classifying a Triangle by Side Lengths
36 in.
Classify the triangle by the lengths of its sides.
All three sides of the triangle are congruent,
so the triangle is both isosceles and equilateral.
Your turn now
36 in.
Classify the triangle by its angle measures.
3.
4.
45
35
5.
80
53
100
60
36 in.
37
40
Classify the triangle by the lengths of its sides.
6.
7.
3 ft
2 ft
8.
18 cm
15 cm
6 cm
7m
7m
7m
2 ft
486
Chapter 10
Geometric Figures
Page 4 of 6
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 714
Getting Ready to Practice
Vocabulary In Exercises 1 and 2, tell whether the statement is
true or false. Explain your reasoning.
1. A triangle that has one angle whose measure is greater than 90 is
called an acute triangle.
2. An equilateral triangle can also be classified as an isosceles triangle.
3. The measures of two of the angles in a triangle are 90 and 57. Find
the measure of the third angle.
4. Camera Tripod A tripod is used to keep a camera
steady while taking pictures. A tripod has three
legs that are each connected to a vertical shaft by
a support bar. Find the value of y.
20
65
95
Classify the triangle by its angle measures.
5. 68, 22, 90
y
6. 53, 57, 70
Classify the triangle by the lengths of its sides.
7. 10 m, 12 m, 16 m
8. 12 in., 30 in., 30 in.
Practice and Problem Solving
with
Example
1
2
3
4
Homework
Exercises
9–11, 13–15
16–19
20–22
23–26
Online Resources
Algebra Find the value of x.
9.
10.
11.
60
42
42
x
58
120
x
33
x
12. Find the Error Describe and correct the error in finding the value of x.
CLASSZONE.COM
• More Examples
• eTutorial Plus
x 56 180
x
56
x 124
The measures of two of the angles in a triangle are given. Find the
measure of the third angle.
13. 112 and 34.8
14. 55.5 and 90.2
15. 128.7 and 30.4
Lesson 10.3
Triangles
487
Page 5 of 6
Algebra Find the value of y.
16.
17.
18.
92
y 44 44
y
y
110
68
40
30
22
19. Drawbridge A castle has a drawbridge
that can be raised or lowered. When
the drawbridge is lowered, what angle
does the chain form with the road
leading up to the castle?
Design
45
45
y
Classify the triangle by its angle measures.
20. 68, 22, 90
21. 82, 64, 34
22. 135, 24, 21
Classify the triangle by the lengths of its sides.
23.
24.
12 cm
■
6 cm
6 cm
6 cm
6 cm
Drawbridge
The Lasalle Street Bridge in
Chicago is a drawbridge on
the Chicago River. The
drawbridge can be raised
70. When fully opened,
what angle does the
drawbridge form with the
road leading up to the
bridge?
25.
11 cm
6 cm
11 cm
11 cm
26. Research Triangle Park Duke University, the University of North
Carolina, and North Carolina State University make up the three
vertices of Research Triangle Park in North Carolina. The distances
between the three universities are 9 miles, 21 miles, and 23 miles.
Classify Research Triangle Park by the lengths of its sides.
Algebra Find the values of x and y.
27.
28.
52
29.
y
x
y
32
x
60
40
32
50
x y
30. Post and Beam In some houses,
wall posts and ceiling beams are
connected by support braces. Find
the unknown angle measures in the
post-and-beam design shown.
30
x
y
31. Challenge The measure of one
angle in a triangle is 36. The other
two angles are congruent. What are
their measures? Explain your
reasoning.
488
Chapter 10
Geometric Figures
Wall
post
e
rac
tb
r
o
p
Sup
Ceiling
beam
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Mixed Review
Order the numbers from least to greatest. (Lesson 4.7)
3
7
32. , 0.42, , 0.4
7
2
3 2 27
33. 6.64, 6 , 6 , 5 3 4
2
5
34. 3.25, 3 , 3.1, 3 3
7
35. Choose a Strategy Use a strategy from the list to solve the
following problem. Explain your choice of strategy.
You buy a prepaid phone card with a
fixed number of minutes on it. You use
one half of your minutes during the
weekend and 6 minutes on Monday. If
24 minutes are left on your phone card,
how many minutes did you start with?
Problem Solving Strategies
Draw a Diagram
Write an Equation
■ Work Backward
■
■
Basic Skills Use a straightedge and a compass to draw a
segment whose length is the sum of the lengths of the two
given segments.
36.
37.
Test-Taking Practice
INTERNET
State Test Practice
38. Multiple Choice Which word describes a triangle whose angle
measures are 42, 58, and 80?
CLASSZONE.COM
A. Right
B. Obtuse
C. Equilateral
D. Acute
39. Short Response A triangle has two angles that measure 53 and 65.
Find the measure of the third angle. Explain your reasoning.
The Shape of Things
Draw a square with a side length of 4 inches on a piece of
paper. Divide the square into 16 smaller squares with side
length 1 inch. Then mark off the lines shown in blue on the
diagram. Cut carefully along these lines to produce the seven
pieces of a tangram.
Arrange all seven tangram pieces
to form each figure.
Lesson 10.3
Triangles
489