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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

Lesson 2

Line Segments and Angles

measuring line segments
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.
slide5
We denote the length of by
  • So, if the line segment below measures 5 inches, then we write
  • We never write

B

A

congruent line segments
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
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

example
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

midpoints
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

example10
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

example11
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

measuring angles
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.
slide15
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.
congruent angles
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.
types of angles
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.

right

obtuse

acute

slide19
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

adjacent angles

1

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
Angle Addition
  • If are adjacent as in the figure below, then

C

A

D

B

example22

M

A

H

T

Example
  • In the figure, is three times and
  • Find
  • Let Then
  • By angle addition,
angle bisectors

A

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.
complementary angles

A

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

example25
Example
  • Find the complement of
  • Call the complement x.
  • Then
example26
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
supplementary angles
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.

1

2

example28
Example
  • Find the supplement of
  • Call the supplement x.
  • Then
example29
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
perpendicular lines
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.

2

1

4

3

slide31
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

m

perpendicular

bisector

n

A

B

vertical angles
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.

3

2

1

4

slide33

2

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.