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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

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).
- We usually place the “zero point” of the ruler at one endpoint and read off the measurement at the other endpoint.

- 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

- 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

- 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

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

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

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

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

- 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.

- 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 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 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

- 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

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 the smaller the angle measurement.

- If are adjacent as in the figure below, then

C

A

D

B

M the smaller the angle measurement.

A

H

T

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

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.

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

Example the smaller the angle measurement.

- Find the complement of
- Call the complement x.
- Then

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 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

Example the smaller the angle measurement.

- Find the supplement of
- Call the supplement x.
- Then

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

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

- 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 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

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.

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