Geometry: Lesson 2: Line Segments and Angles

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Lesson 2
Line Segments and Angles
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Measuring Line Segments
• The instrument used to measure a line
segment is a scaled straightedge like a
ruler or meter stick.
• Units used for the length of a line segment
include inches (in), feet (ft), centimeters
(cm), and meters (m).
• We usually place the “zero point” of the
ruler at one endpoint and read off the
measurement at the other endpoint.
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Rulers
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• We denote the length of AB by AB
• So, if the line segment below measures 5
inches, then we write
• We never write
AB  5 in
AB  5 in
A
B
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Congruent Line Segments
• In geometry, two figures are said to be
congruent if one can be placed exactly on
top of the other for a perfect match. The
symbol for congruence is  .
• Two line segments are congruent if and
only if they have the same length.
• So, AB  CD if and only if AB  CD.
• The two line segments below are
congruent.
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Segment Addition
• If three points A, B, and C all lie on the
same line, we call the points collinear.
• If A, B, and C are collinear and B is
between A and C, we write A-B-C.
• If A-B-C, then AB+BC=AC. This is known
as segment addition and is illustrated in
the figure below.
A
B
C
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Example
R
• In the figure, suppose RS = 7
and RT = 10. What is ST?
• We know that RS + ST = RT.
• So, subtracting RS from both
sides gives:
ST  RT  RS
 10  7
3
S
T
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Midpoints
• Consider CD on the right.
• The midpoint of this segment is a
point M such that CM = MD.
• M is a good letter to use for a
midpoint, but any letter can be
used.
C
M
D
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Example
• In the figure, it is given that B is the
midpoint of AC and D is the midpoint of CE.
• It is also given that AC = 13 and DE = 4.5.
Find BD.
• Note that BC is half of AC. So, BC =
0.5(13) = 6.5.
• Note that CD equals DE. So, CD = 4.5.
• Using segment addition, we find that BD =
BC + CD = 6.5 + 4.5 = 11.
A
B
C
D
E
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Example
P
• In the figure T is the midpoint of
PQ.
• If PT = 2(x – 5) and TQ = 5x – 28,
then find PQ.
• We set PT and TQ equal and
solve for x:
T
Q
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Example continued
2( x  5)  5 x  28
distribute: 2 x  10  5 x  28
subtract 2 x :
10  3 x  28
add 28 :
divide by 3:
18  3x
6x
Now PT  2(6  5)  2(1)  1 and
TQ  5(6)  28  30  28  2.
So, PQ  PT  TQ  2  2  4.
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Measuring Angles
• Angles are measured using a protractor, which
looks like a half-circle with markings around its
edges.
• Angles are measured in units called degrees
(sometimes minutes and seconds are used too).
• 45 degrees, for example, is symbolized like this:
45
• Every angle measures more than 0 degrees and
less than or equal to 180 degrees.
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A Protractor
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• The smaller the opening between the two
sides of an angle, the smaller the angle
measurement.
• The largest angle measurement (180
degrees) occurs when the two sides of the
angle are pointing in opposite directions.
• To denote the measure of an angle we
write an “m” in front of the symbol for the
angle.
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• Here are some common angles and their
measurements.
m1  45
1
m2  90
2
m3  135
3
4
m4  180
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Congruent Angles
• Remember: two geometric figures are
congruent if one can be placed exactly on
top of the other for a perfect match.
• So, two angles are congruent if and only if
they have the same measure.
• So, A  B if and only if mA  mB.
• The angles below are congruent.
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Types of Angles
• An acute angle is an angle that measures less
than 90 degrees.
• A right angle is an angle that measures exactly
90 degrees.
• An obtuse angle is an angle that measures
more than 90 degrees.
acute
right
obtuse
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• A straight angle is an angle that measures
180 degrees. (It is the same as a line.)
• When drawing a right angle we often mark
its opening as in the picture below.
right angle
straight angle
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Adjacent Angles
• Two angles are called adjacent angles if
they share a vertex and a common side
(but neither is inside the opening of the
other).
• Angles 1 and 2 are adjacent:
1 2
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Angle Addition
• If ABC and CBD are adjacent as in
the figure below, then
mABC  mCBD  mABD
C
A
D
B
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Example
• In the figure, mMAH is three
times mHAT and mMAT  132.
• Find mMAH .
A
• Let mHAT  x. Then
M
mMAH  3x.
• By angle addition,
x  3x  132
4 x  132
x  33
So, mMAH  3  33  99.
H
T
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Angle Bisectors
• Consider ABC below.
• The angle bisector of this angle is the ray
BD such that mABD  mDBC.
• In other words, it is the ray that divides the
angle into two congruent angles.
A
D
B
C
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Complementary Angles
• Two angles are complementary if their
measures add up to 90.
• If two angles are complementary, then each
angle is called the complement of the other.
• If two adjacent angles together form a right
angle as below, then they are
complementary.
A
1 and 2 are
complementary
B
if ABC is a
right angle 24
1
2 C
Example
• Find the complement of 37.
• Call the complement x.
• Then
37  x  90
x  90  37
x  53
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Example
•
•
•
•
Two angles are complementary.
The angle measures are in the ratio 7:8.
Find the measure of each angle.
The angle measures can be represented
by 7x and 8x. Then
7 x  8 x  90
15 x  90
x  6
Then the angle measures are
7 x  42 and 8 x  48.
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Supplementary Angles
• Two angles are supplementary if their
measures add up to 180.
• If two angles are supplementary each
angle is the supplement of the other.
• If two adjacent angles together form a
straight angle as below, then they are
supplementary.
1
2
1 and 2 are
supplementary
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Example
• Find the supplement of 62.
• Call the supplement x.
• Then
62  x  180
x  180  62
x  118
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Example
• One angle is 30 more than twice another angle.
If the two angles are supplementary, find the
measure of the smaller angle.
• Let x represent the measure of the smaller angle.
Then 2 x  30 represents the measure of the
larger angle. Then
x  (2 x  30)  180
3x  30  180
combine like terms:
subtract 30 :
3x  150
x  50
divide by 3:
So the smaller angle measures 50.
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Perpendicular Lines
• Two lines are perpendicular if
they intersect to form a right
angle. See the diagram.
• Suppose angle 2 is the right
angle. Then since angles 1
and 2 are supplementary,
angle 1 is a right angle too.
Similarly, angles 3 and 4 are
right angles.
• So, perpendicular lines
intersect to form four right
angles.
1
3 4
2
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• The symbol for perpendicularity is  .
• So, if lines m and n are perpendicular, then we
write m  n.
• The perpendicular bisector of a line segment is
the line that is perpendicular to the segment and
that passes through its midpoint.
m
m
perpendicular
mn
bisector
n
A
B
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Vertical Angles
• Vertical angles are two angles that are formed
from two intersecting lines. They share a vertex
but they do not share a side.
• Angles 1 and 2 below are vertical.
• Angles 3 and 4 below are vertical.
1
3
2
4
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• The key fact about vertical angles is that they
are congruent.
• For example, let’s explain why angles 1 and 3
below are congruent. Since angles 1 and 2 form
a straight angle, they are supplementary. So,
m1  180  m2.
• Likewise, angles 2 and 3 are supplementary.
So, m3  180  m2. So, angles 1 and 3 have
the same measure and they’re congruent.
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2
3
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