Download Lines and Angles

7.4
Lines and Angles
7.4
OBJECTIVES
1.
2.
3.
4.
NOTE “Geo” means earth, just
as it does in the words
“geography” and “geology.”
Distinguish between lines and line segments
Determine when lines are perpendicular or parallel
Determine whether an angle is right, acute, or obtuse
Use a protractor to measure an angle
After counting, there was geometry. Once the Egyptians and Babylonians had mastered the
counting of their animals, they became interested in measuring their land. This is the foundation of geometry. Literally translated, “geometry” means earth measurement. Many of
the topics we consider in geometry (topics such as angles, perimeter, and area) were first
studied as part of surveying.
As is usually the case, we start the study of a new topic by learning some vocabulary.
Most of the terms we will discuss will be familiar to you. It is important that you understand what we mean when we use these words in the context of geometry.
We begin with the word “point.” A point is a location; it has no size and covers no area.
If we string points together forever, we create a line. In our studies we will consider only
straight lines. We use arrowheads to indicate that a line goes on forever.
A piece of a line that has two endpoints is called a line segment.
Example 1
Recognizing Lines and Line Segments
NOTE The capital letters are
Label each of the following as a line or a line segment.
labels for points.
C
A
E
B
F
D
(c)
(b)
(a)
Both a and c continue forever in both directions. They are lines. Part b has two endpoints.
It is a line segment.
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CHECK YOURSELF 1
Label each of the following as a line or a line segment.
F
B
C
A
(1)
E
(2)
(3)
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CHAPTER 7
GEOMETRY AND MEASURE
Definitions: Angle
An angle is a geometric figure consisting of two line segments that share a
common endpoint.
A
O
B
OA and OB are line segments. O is the vertex of the angle.
Surveyors use an instrument called a transit. A transit allows surveyors to measure angles
so that, from a mathematical description, they can determine exactly where a property line is.
Definitions: Perpendicular Lines
When two lines cross ( or intersect) they form four angles. If the lines intersect
such that four equal angles are formed, we say that the two lines are
perpendicular.
At most highway intersections, the two roads are perpendicular.
Definitions: Parallel Lines
If two lines are drawn so that they never intersect (even if we extend the lines
forever), we say that the two lines are parallel.
Parallel parking gets its name from the fact that the parking spot is parallel to the traffic lane.
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LINES AND ANGLES
SECTION 7.4
577
Example 2
Recognizing Parallel and Perpendicular Lines
Label each pair of lines as parallel, perpendicular, or neither.
(a)
(c)
(b)
Although we don’t see the lines in part a intersecting, if they were extended as the arrowheads indicate, they would. The lines of part b are perpendicular because the four angles
formed are equal. Only the lines in part c are parallel.
CHECK YOURSELF 2
Label each pair of lines as parallel, perpendicular, or neither.
(1)
NOTE You may recall seeing
this small square in Chapter 4.
There we used it to show the
altitude (height) of a triangle.
(3)
(2)
We call the angle formed by two perpendicular lines or line segments a right angle. We
designate a right angle by forming a small square.
A
B
O
We can refer to a specific angle by naming three points. The middle point is the vertex
of the angle.
Example 3
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Naming an Angle
Name the highlighted angle.
B
A
C
O
D
NOTE We could also call this
angle BOC.
The vertex of the angle is O, and the angle begins at C and ends at B, so we would name
the angle COB.
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CHAPTER 7
GEOMETRY AND MEASURE
CHECK YOURSELF 3
Name the highlighted angle.
B
C
O
A
D
One way to measure an angle is to use a unit that we call a degree. There are 360 degrees
(we write this as 360°) in a complete circle. Note in the picture on the left that there are four
right angles in a circle. If we divide 360° by 4, we find that each right angle must measure
90°. Here are some other angles with their measurements.
120
60
30
180
An acute angle measures between 0° and 90°. An obtuse angle measures between 90°
and 180°. A straight angle measures 180°.
Example 4
Labeling Types of Angles
Label each of the following angles as an acute, obtuse, right, or straight angle.
(a)
(b)
(c)
(d)
Part a is obtuse (the angle is more than 90°). Part b is a right angle (designated by the
small square). Part c is an acute angle (it is less than 90°), and part d is a straight angle.
Label each angle as an acute, an obtuse, a right, or a straight angle.
(1)
(2)
(3)
(4)
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CHECK YOURSELF 4
LINES AND ANGLES
SECTION 7.4
579
When assigning a measurement to an angle, we usually use a tool called a protractor.
NOTE Your protractor may
show the degree measures in
both directions.
100
110
80
70
90 80
70
90 100 1
60
10
12
0
50
13
0
0
10
20
0 180
30 160 17
0
15
40
0
14
180 170
160
0 10
15
0
20
14
30
0
40
0
12
0
60
13
50
Place the protractor so that the
vertex of the angle is here.
We read the protractor by placing one line segment of the angle at 0°. We then read the
number that the other line segment passes through. This number represents the degree measurement of the angle. The point at the center of the protractor, the endpoint of the two line
segments, is the vertex of the angle.
Example 5
Measuring an Angle
Use the protractor to estimate the measurement for each angle.
F
B
D
A
O
C
E
O
O
The measure of AOB is 45°. The measure of COD is 150°. The measure of EOF is
between 50° and 55°. We could estimate that it is a 52° angle.
CHECK YOURSELF 5
Use a protractor to estimate the measurement for each angle.
D
© 2001 McGraw-Hill Companies
B
A
O
(2)
F
E
O
(3)
C
O
(1)
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CHAPTER 7
GEOMETRY AND MEASURE
If we wish to refer to the degree measure of ABC, we use mABC.
Example 6
Measuring an Angle
Find mAOB.
B
C
A
O
D
NOTE mAOB 20° is read
Using the protractor, we find mAOB 20°.
“the measure of angle AOB is
20 degrees.”
CHECK YOURSELF 6
Find mAOC.
C
B
D
A
O
E
F
CHECK YOURSELF ANSWERS
(1) Line segment; (2) line segment; (3) line
(1) Parallel; (2) neither; (3) perpendicular
(1) Right; (2) straight; (3) acute; (4) obtuse
(1) 120°; (2) 80°; (3) 160°
6. 135°
3. BOA or AOB
© 2001 McGraw-Hill Companies
1.
2.
4.
5.
Name
Exercises
7.4
Section
1. Draw line segment AB.
#
A
2. Draw line EF.
#
B
#
E
3. Draw line AC.
#
A
Date
#
F
ANSWERS
1.
4. Draw line segment BC.
#
C
#
B
#
C
2.
3.
Identify each object as a line or line segment.
4.
5.
6.
7.
P
D
5.
U
6.
O
C
V
7.
8.
8.
9.
10.
A
X
K
9.
10.
L
W
11.
12.
B
13.
11.
12.
H
E
14.
15.
F
G
16.
17.
Label exercises 13 to 18 as true or false.
18.
13. There are exactly two different line segments that can be drawn through two points.
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14. There are exactly two different lines that can be drawn through two points.
15. Two opposite sides of a square are parallel line segments.
16. Two adjacent sides of a square are perpendicular line segments.
17. ABC will always have the same measure as CAB.
18. Two acute angles have the same measure.
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ANSWERS
19.
19. Are the following two lines parallel, perpendicular, or neither?
20.
21.
22.
23.
20. Are the following two lines parallel, perpendicular, or neither?
24.
25.
26.
27.
28.
29.
Give an appropriate name for each indicated angle.
21.
30.
22.
Q
P
23.
U
M
V
R
31.
32.
N
S
O
L
T
24.
25.
A
F
26.
G
X
Y
Z
H
E
B
W
C
27.
28.
S
J
K
R
L
I
V
N
U
M
For each angle described, give its measure in degrees. One revolution is a full circle. Sketch
the angle.
582
29. A represents
1
of a revolution
6
30. B represents
1
of a revolution
3
31. C represents
7
of a revolution
12
32. D represents
11
of a revolution
12
© 2001 McGraw-Hill Companies
T
ANSWERS
33.
Measure each angle with a protractor. Identify the angle as acute, right, obtuse, or
straight.
33.
34.
A
34.
E
35.
D
36.
37.
B
O
F
38.
39.
35.
36.
O
P
40.
D
C
R
Q
37.
38.
F
G
O
E
D
F
In the figure, two parallel lines are intersected by a third line, forming eight angles. Draw
lines like these on your paper.
1
© 2001 McGraw-Hill Companies
3
5
7
2
4
6
8
39. Use your protractor to measure 2 and 6. What do you notice?
40. Use your protractor to measure 3 and 6. What do you notice?
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ANSWERS
41. Draw any triangle using a ruler. With your protractor, carefully measure the three
41.
interior angles, and find their sum. Do this again with two more triangles of different
shapes. What do you notice about the sums of the angles? Make a conjecture about
the sum of the angles of any triangle.
42.
42. A quadrilateral is a four-sided polygon. Draw any quadrilateral, and measure the
43.
four interior angles with a protractor. Record these, and find their sum. Make a
conjecture concerning the sum of the interior angles of any quadrilateral. Test your
conjecture on another quadrilateral.
44.
43. A pentagon is a five-sided polygon. Draw any pentagon, and measure the five
interior angles with a protractor. Record these, and find their sum. Make a conjecture
concerning the sum of the interior angles of any pentagon. Test your conjecture on
another pentagon.
44. A hexagon is a six-sided polygon. Draw any hexagon, and measure the six interior
angles with a protractor. Record these, and find their sum. Make a conjecture
concerning the sum of the interior angles of any hexagon. Test your conjecture on
another hexagon.
Answers
1.
A
B
3.
A
C
5. Line
© 2001 McGraw-Hill Companies
7. Line segment
9. Line segment
11. Line
13. False
15. True
17. False
19. Parallel
21. POQ
23. MNL
25. FEG
27. SVT
29. 60°
31. 210°
33. 135°; obtuse
35. 90°; right
37. 30°; acute
39. 40°; 40°
41.
43.
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