1 Lesson 2 Line Segments and Angles. 2 3 Measuring Line Segments The instrument used to measure a line segment is a scaled straightedge like a ruler.

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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