Measuring Angles and Segments

Lesson Plan - Get It!

Audio:

Below is a list of vocabulary words that you will need to understand in order to work successfully in this lesson.

• postulate: a simple and direct statement that can easily be accepted as true without prof
• congruent: exactly the same in size and shape
• angle: the common endpoint shared by two rays
• acute angle: an angle that measures less than 90°
• right angle: an angle that measures 90°
• obtuse angle: an angle that measures more than 90° but less than 180°
• straight angle: an angle that measures 180°

Now, review four geometric postulates about line segment and angle measurement. Each will be explored in greater detail in this lesson:

Ruler Postulate

The points of a line can be put into a one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers.

Segment Addition Postulate

If three points A, B, and C are collinear, and B is between A and C, then AB + BC = AC.

Protractor Postulate

Let ray OA and ray OB be opposite rays in a plane.

Ray OA, ray OB, and all the rays with endpoint O that can be drawn on one side of line AB can be paired with the real numbers from 0 to 180 in such a way that:

• ray OA is paired with 0 and ray OB is paired with 180
• If ray OC is paired with x and ray OD is paired with y, then the measurement of angle COD equals the absolute value of x - y.

Angle Addition Postulate

If point B is in the interior of angle AOC, then the measurement of angle AOB + the measurement of angle BOC equals the measurement of angle AOC.

If angle AOC is a straight angle, then the measurement of angle AOB + the measurement of angle BOC equals 180°.

The postulates listed above can sound confusing. Let's take a look at them one at a time to make them easier to understand.

Ruler Postulate

The ruler postulate explains that the measurement of a line segment can be found by subtracting the values of the two endpoints and finding the absolute value of that answer.

For example, below is a line segment:

To find the measurement of this line segment, you need to measure it with a number line:

It doesn't matter where you put the line segment on the number line; just line it up so each endpoint corresponds with a number on the number line.

As you can see in this example, the endpoints correspond with -10 and -3.

Subtract the values of the endpoints:

-10 - (-3) = -7

Take the absolute value of the answer:

|-7| = 7

Quick reminder:

A ruler and a protractor are both examples of number lines.

In other words, when presented with a math problem where you are asked to find the measurement of a line segment, it is not necessary for you to draw a number line under the line segment. You can use a ruler or protractor to complete this task.

Segment Addition Postulate

The segment addition postulate explains that if there are 3 points on the same line (collinear), then the measurement between the first two points plus the measurement between the last two points is the same as the measurement between the first and third points.

In the example above, notice that there are three collinear points (A, B, and C).

The segment addition postulate states that if you find the measurement of segment AB and the measurement of segment BC and add them together, you will get the measurement of segment AC.

The value of point A is -10, and the value of point B is -3. The measurement of segment AB is:

|-10 - (-3)| = |-7| = 7

The value of point B is -3, and the value of point C is 3. The measurement of segment BC is:

|-3 - 3| = |-6| = 6

So the measurement of segment AC is:

7 + 6 = 13

You know this is true when you use the ruler postulate to measure the segment AC.

The value of point A is -10, and the value of point C is 3. The measurement of segment AC is:

|-10 - 3| = |-13| = 13

Protractor Postulate

The protractor postulate states that two rays that are going in opposite directions but share the same endpoint — and are drawn on the same side of a line — can be measured between the numbers 0 and 180:

This postulate further states that if two more rays, sharing O as their endpoint, are drawn into the above illustration; you can find the measurement of the angle they make by subtracting the value of one ray minus the value of the second ray:

Look at the illustration above.

The value of ray OC is 50, and the value of ray OD is 120. Also notice that both rays share the same endpoint, O, which constructs an angle, COD.

To find the measurement of angle COD, simply subtract the value of ray OC from the value of ray OD:

120 - 50 = 70

Therefore, the measurement of angle COD is 70°.

Angle Addition Postulate

The angle addition postulate states that you can locate a point inside of an angle and draw a ray that splits the original angle into two angles.

It also states that when you add the measurement of each of the two angles together, you will get the measurement of the original angle:

Once you draw a ray connecting points O and C, you can then use a protractor to measure angle AOC and angle COB.

You would then add those measurements together to get the measurement of angle AOB.

• Why do you need to know the postulates discussed above?

That is a good question!

You should always try to relate math concepts to real-world examples so you can better understand them and know when to use them.

In the real world, you know that when you use a ruler to measure an object, each point of the object coincides with only one number on the ruler (ruler postulate).

Knowing this, you can now measure two objects and add their measurements together to find the combined measurement.

For example, assume you have a stone patio in the back of your home. Assume you want to extend that stone patio, and you want to cover it with wood.

You measure the length of the stone patio and find that it is 4 feet long.

You place a marker where you want the length of the new patio to end.

Now you measure from the end of the stone patio to the marker and find that distance is also 4 feet.

At this point, you know you need to prepare that second 4-foot area to become a patio, and you also know you need 8-foot-long pieces of wood to make your patio.

This is an example of  the segment addition postulate.

• Have you ever played a game of pool?

The goal of this game is to get all your balls into pockets located around the table.

Assume your pool ball is close to one of the corner pockets.

You can use the protractor postulate and angle addition postulate to calculate the angle at which you will need to hold your pool cue to hit the ball and assure that it lands in the pocket.

The next section contains activities to help you practice the four postulates you learned about in this lesson. Enjoy!