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Lesson 2Line 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 by
• So, if the line segment below measures 5 inches, then we write
• We never write
AB AB
5 inAB
5 inAB
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, • The two line segments below are
congruent.
.
if and only if .AB CD AB CD
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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
• 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:
R
S
T
ST RT RS 10 7 3
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Midpoints
• Consider 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.
CD C
D
M
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Example
• In the figure, it is given that B is the midpoint of and D is the midpoint of
• 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
AC .CE
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Example
• In the figure T is the midpoint of
• If PT = 2(x – 5) and TQ = 5x – 28, then find PQ.
• We set PT and TQ equal and solve for x:
P
Q
T
.PQ
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Example continued
2( 5) 5 28x x distribute: 2 10 5 28x x
subtract 2 :x 10 3 28x
add 28 : 18 3xdivide by 3 : 6 x
Now 2(6 5) 2(1) 1 and
5(6) 28 30 28 2.
PT
TQ
So, 2 2 4.PQ PT TQ
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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:
• Every angle measures more than 0 degrees and less than or equal to 180 degrees.
45
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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.
1 2
3
4
1 45m 2 90m
3 135m
4 180m
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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,
• The angles below are congruent.
if and only if .A B m A m B
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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.
straight angle
right 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 are adjacent as in the figure below, then
m ABC
A
B
C
and ABC CBD
D
m CBD m ABD
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Example
• In the figure, is three times and
• Find
• Let Then
• By angle addition,
A
M
TH
m MAHm HAT 132 .m MAT
.m MAH.m HAT x
3 .m MAH x
3 132x x 4 132x
33x So, 3 33 99 .m MAH
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Angle Bisectors
• Consider below. • The angle bisector of this angle is the ray such that• In other words, it is the ray that divides the
angle into two congruent angles.
A
BC
D
ABC
BD��������������
.m ABD m DBC
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Complementary Angles
• Two angles are complementary if their measures add up to
• 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.
90 .
1 2
A
BC
1 and 2 are
complementary
if is a
right angle
ABC
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Example
• Find the complement of
• Call the complement x.
• Then
37 .
37 90x 90 37x 53x
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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 8 90x x 15 90x
6x Then the angle measures are
7 42 and 8 48 .x x
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Supplementary Angles
• Two angles are supplementary if their measures add up to
• 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.
180 .
1 2
1 and 2 are
supplementary
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Example
• Find the supplement of
• Call the supplement x.
• Then
62 .
62 180x 180 62x 118x
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Example
• One angle is 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 represents the measure of the larger angle. Then
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(2 30 ) 180x x
2 30x
combine like terms: 3 30 180x subtract 30 : 3 150x
divide by 3: 50x
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.
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3 4
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• The symbol for perpendicularity is• So, if lines m and n are perpendicular, then we
write • The perpendicular bisector of a line segment is
the line that is perpendicular to the segment and that passes through its midpoint.
.
.m n
m
nm n
m
A B
perpendicular
bisector
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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 23
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,
• Likewise, angles 2 and 3 are supplementary.
So, So, angles 1 and 3 have the same measure and they’re congruent.
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1 180 2.m m
3 180 2.m m