1. Line Segments and Angles
Lesson 2
Line Segments and Angles

3. 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.

4. Rulers

5. We denote the length of by
So, if the line segment below measures 5 inches, then we write
We never write A B

6. 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.

7. 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

8. 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: S T

9. 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. M D

10. 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 = 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 = = 11. B C D E

11. 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: T Q

12. Example continued

13. 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.

14. A Protractor

15. 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.

16. Here are some common angles and their measurements.1 2 4 3

17. 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.

18. 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

19. A straight angle is an angle that measures 180 degrees
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

20. 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

21. Angle Addition
If are adjacent as in the figure below, then C A D B

22. Example In the figure, is three times and Find Let Then
By angle addition, A M T H

23. 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 B C D

24. 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. A B C 1 2

25. Example
Find the complement of
Call the complement x.
Then

26. 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

27. 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

28. Example
Find the supplement of
Call the supplement x.
Then

29. 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

30. 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 3 4

31. 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

32. 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 1 2 4

33. 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. 1 2 3