Unit 1: Section 2: I can determine the measure of complementary, supplementary, linear pair, and vertical angles

UNIT 1

Section 2: I can determine the measure of Complementary, Supplementary, Linear Pair, and Vertical Angles

Welcome to another installment of the class blog! Today, we will be discussing four special kinds of angle pairs. These angle pairs will continue to come up in class the whole year, so it is very important that you absorb and understand the content of this blog. It goes without saying, but you should not be reading this section unless you have already read and understand both the Vocabulary and Section 1.

Now that we've got that out of the way, here are the four special angle pairs.

Complementary Angles

Complementary angles are two angles whose sum is 90 degrees.

No, complementary angles are not something you get for free or some kind of praise. They are a pair of angles whose sum is 90 degrees.

Do they have to be adjacent? No. Do they have to be congruent? No. All they have to do is add up to 90 degrees. This does not mean that one or the other of them is a 90-degree angle (a right angle). The two angles add up to 90 degrees.

Below are some pictures of complementary angles.

Figure 1: Complementary angles that are adjacent.

Figure 2: Complementary angles that are non-adjacent.

What would happen if we didn't know both angles, but we knew they were complementary? Well, it's actually just as easy as our Angle Addition Postulate. Check this out:

Solving Problems With Complementary Angles

Below we have two complementary angles, and we only know the measure of one of them.

This problem can be easily solved using the definition of complementary angles. Remember, these two angles must add up to 90 degrees, and that gives us a big hint in how to begin.

LKJ + GHI = 90.

With a little substitution, this becomes the equation 34 + x = 90, and now it is a very easy one-step equation to determine that the angle marked GHI must be 56 degrees. Piece of cake, right! On to our next special pair.

Supplementary Angles

Supplementary angles are two angles whose sum is 180 degrees.

This one is extremely similar to complementary angles, only the sum is 180 degrees instead of 90. Do they have to be adjacent? No. Do they have to be congruent? No. All they have to do is add up to 180 degrees. This does not mean that one or the other of them is a 180-degree angle (straight angle). The two angles add up to 180 degrees.

Below are some pictures of supplementary angles.

Figure 3: Supplementary angles that are adjacent.

Figure 4: Supplementary angles that are non-adjacent.

What would happen if we didn't know both angles, but we knew they were supplementary? It is just as easy as our other example.

Solving Problems With Supplementary Angles

Below we have two supplementary angles, and we only know the measure of one of them.

This problem can be easily solved using the definition of supplementary angles. Remember, these two angles must add up to 180 degrees, and that gives us a big hint in how to begin.

COB + BOA = 180

With a little substitution, this becomes the equation x + 38 = 180, and now it is a very easy one-step equation to determine that the angle marked COB must be 142 degrees. Man, that was almost easier than the first one! Let's move on to the next pair. Although I should let you know, this next pair is just a special type of what we just learned about, supplementary angles.

Linear Pairs

A linear pair is a pair of adjacent angles whose sum is 180 degrees.

This one is a special kind of supplementary angles. In fact, if you look back at Figure 3, there is an example of this pair of angles. Linear pairs meet two requirements. 1) They are adjacent, meaning they share a side, and 2) they are supplementary, meaning their measures add up to 180 degrees. Here is a Venn Diagram to help you out.

Figure 5: You can see from the Venn Diagram that Linear Pairs are

both adjacent and supplementary

As you might have guessed at this point, there can be problems with linear pairs where you have one angle and not the other, and that works out exactly like in the supplementary angles section.

Sometimes, though, you'll have to work with neither of the angles, but I'll give you the relationship between the two, and you will have to work from there. Let me show you an example.

Solving Problems With Angle Relationships

Imagine that I told you I knew of two angles that formed a linear pair, and that one of them was twice as big as the other, but I knew nothing else about them. Is this enough information to figure out both angles?

It turns out that yes, it is! Here's how it works. We start with what we know.

The angles are a linear pair. That means that, whatever the angles are (say angle A and angle B):

A + B = 180                                                    (1)

We also know that one angle is twice as big as the other. Believe it or not, this gives us a lot of help.

Let's assume the big angle is angle B. What is its relationship to angle A? Well, it is twice as big, so whatever measure angle A is, angle B has to be twice that big. This means that

B = 2A                                                    (2)

We can substitute this value into our first line of work, and it will give us that

A + 2A = 180.                                                 (3)

All we have to do is combine like terms to give us

3A = 180                                                 (4)

and then the tiniest bit of Algebra will give us

A = 60.                                                  (5)

This work gives us the measure of angle A. Awesome!

"But, Mr. Shock," you're saying to your screen - as though I can hear you from the comfort of my home - "that's only one angle. You said we'd be able to solve both of them!"

So I did, and I keep my word. All we have to do is look back at the relationship we established in our second line, and we can see that whatever A ends up being, B is going to be double that!. So since A is 60, then

B = 2A                                                    

B = 2(60)                                                

B = 120                                              (6)

These problems can look tough in the beginning, but with enough practice, they too will be as easy as your multiplication facts (or times tables, as some of you call them). So, make sure that you practice all of these special angles at home as well as in class. Let's move on to our last pair of the day.

Vertical Angles

Vertical angles are two congruent opposite angles formed by intersecting lines.

These pairs are different than the other pairs we've seen today in a few ways. The first is that they are congruent. This means that they are the exact same size, or that they have the same measure. The second is that vertical angles are never adjacent. Some examples are shown below.

Figure 6: The two red angles and the two blue angles are vertical angles.

When a problem is based on vertical angles, the equations write themselves, because one angle is equal to the other.

Solving Problems With Vertical Angles

In this problem, the first question we might be asked is "what is the value of y?" This question is extremely easy once we apply vertical angles. There are two pairs of vertical angles here, 76 & y, and x & z. Like I said above, the equations write themselves since vertical angles are congruent.

76 = y

and

x = z.

And we know that angle y is 76 degrees. Boom! Done.

Because of the ease of problems with vertical angles, we often also ask you to find angle x and angle z. This might seem like it's not possible, but if we remember how linear pairs work, it becomes much clearer how to solve it. Looking at the picture above, we can see that x and y form a linear pair, and also that y and z form another linear pair. This will give us two equations.

x + y = 180

and

y + z = 180

This, paired with the fact that y is 76 degrees gives us

x + 76 = 180

and

76 + z = 180

And again, using the tiniest bit of Algebra, we can find that both x and z are 104 degrees. Holy cow! They're both the same!

Of course they're the same. If you remember a few paragraphs ago, we said that x and z were a pair of vertical angles, so they must be the same.

Summary

In closing, let's go over the four angle pairs we just read about.

complementary: two angles whose measures add to 90. A + B = 90

supplementary: two angles whose measures add to 180. A + B = 180

linear pair: two adjacent angles whose measures add to 180. A + B = 180

vertical angles: two congruent angles on opposite sides of intersecting lines. A = B, C = D

Attached Work

Don't forget to practice your vertical angles, complementary and supplementary angles, and take the attached quiz before moving on.