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Lesson 2. Line Segments and Angles. 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).

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Lesson 2 l.jpg

Lesson 2

Line Segments and Angles


Measuring line segments l.jpg
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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B

A


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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.


Segment addition l.jpg
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:

S

T


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Midpoints

C

  • 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.

M

D


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Example

A

  • 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.

B

C

D

E


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Example

P

  • 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:

T

Q



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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.



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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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1

2

4

3


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Congruent Angles the smaller the angle measurement.

  • 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.


Types of angles l.jpg
Types of Angles the smaller the angle measurement.

  • 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.

right

obtuse

acute


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  • A the smaller the angle measurement. 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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1 the smaller the angle measurement.

2

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:


Angle addition l.jpg
Angle Addition the smaller the angle measurement.

  • If are adjacent as in the figure below, then

C

A

D

B


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M the smaller the angle measurement.

A

H

T

Example

  • In the figure, is three times and

  • Find

  • Let Then

  • By angle addition,


Angle bisectors l.jpg

A the smaller the angle measurement.

D

C

B

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.


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A the smaller the angle measurement.

C

B

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.

1

2


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Example the smaller the angle measurement.

  • Find the complement of

  • Call the complement x.

  • Then


Example26 l.jpg
Example the smaller the angle measurement.

  • 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


Supplementary angles l.jpg
Supplementary Angles the smaller the angle measurement.

  • 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.

1

2


Example28 l.jpg
Example the smaller the angle measurement.

  • Find the supplement of

  • Call the supplement x.

  • Then


Example29 l.jpg
Example the smaller the angle measurement.

  • 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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Perpendicular Lines the smaller the angle measurement.

  • 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.

2

1

4

3


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  • The symbol for perpendicularity is the smaller the angle measurement.

  • 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

m

perpendicular

bisector

n

A

B


Vertical angles l.jpg
Vertical Angles the smaller the angle measurement.

  • 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.

3

2

1

4


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2 the smaller the angle measurement.

1

3

  • 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.