SUPPLEMENTARY, COMPLIMENTARY, AND VERTICAL ANGLES

OBJECTIVE

Go over what an Angle is

Go over Complimentary Angles

Go over Supplementary angles

Go over Linear Pairs

Go over Vertical Angles

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So what is an angle?

So, according to all knowing wiki:

“In Euclidean geometry (which is what we’re studying), an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.”

Or in plain English, an angle is simply what results when two lines (or rays) intersect.

Now, before we can continue, we have a definitions to go over:

Intersect – things intersect when they cross over each other. For example:

As we can see here, line AB intersects line CB at point E.

So, if we look at this example, we can see that there are actually 4 angles formed from this intersection.

We have:

Different types of angles

So now that we know what an angle actually is, let’s look over what different types of angles there are.

Now, this may seem tedious and ridiculous, but we give them names so we can tell the difference between different angles.

Basically, giving an angle a name is easier than describing it (you’ll see what is meant in the next slide.)

Complimentary Angles

Complimentary angles are angles that add to 90 degrees.

That’s it.

If two angles add up to 90 degrees, then they are complimentary angles.

Like so:

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So why are they called complimentary angles instead of something else?

It goes back to Greek.

“Completum” means “completed”, and the idea was that a right angle was a completed angle.

Mainly because the Greeks used it a TON for building durable buildings.

(Which we still do to this day).

Now there are a few types of Complimentary angles.

As we can see, these two angles are complimentary because if we add them together:

We get 90 degrees

Types of Complimentary Angles

So there are two different types of complimentary angles

(Or any angles for that matter)

Adjacent, and Non-Adjacent

Adjacent angles are just angles that are connected right next to each other.

Non-Adjacent angles aren’t connected.

So, adjacent complimentary angles are something like:

As you can see, these two angles are literally right next to each other.

Or, another way you can see this, they both share a side.

So if adjacent complementary angles are right next to each other

Then non-adjacent complementary angles are something like:

As you can see, these two angles are no where close to each other.

They both have their own sides, but still add up to 90 degrees.

So they are complimentary, just non-adjacent.

SO WHAT ABOUT SUPPLEMENTARY ANGLES?

Supplementary angles are angles that add up to 180 degrees.

An easy way to remember is:

C comes before S in the alphabet, so C is for the smaller angle (90 degrees)

And S is for the larger angle (180 degrees).

Again, these angles can be adjacent, or non adjacent.

So for example, an adjacent supplementary angle would be:

As you can see, these angles add up to 180, and they are right next to each other.

(Or they share a side)

And a pair of non-adjacent supplementary angles would be something like:

Again, these angles are not right next to each other, so they aren’t considered adjacent.

HOWEVER…..

Adjacent supplementary angles actually have a name, and that name is:

Linear Pairs.

So basically, two angles are considered a Linear Pair if:

They are supplementary and

They are adjacent.

But don’t take my word for it, here’s something to show you that it’s true.

(Welcome to your first proof).

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LINEAR PAIR THEOREM

SO, WHAT DOES THIS MEAN?

This means that any angles that make up a linear pair are supplementary.

This essentially makes it a lot easier to work with when we are dealing with angles that we aren’t sure what they add up to.

Example

So, here’s an example of a problem you may encounter:

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SO, BASICALLY WE’RE ADDING THEM TOGETHER TO GET 180

Once we find out the two angles are linear pairs, that is exactly what we’re doing.

This is why it’s important to start a journal to log all of the definitions and theorems in.

It will come in handy later on (like, I dunno, maybe come test time?)

LAST ANGLE FOR TODAY: VERTICAL ANGLES

So, now we know all about complementary angles, supplementary angles, and linear pairs

But what about vertical angles?

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Vertical angles, by definition, are a pair of non-adjacent angles formed when two lines intersect.

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So, in plain English, basically when two lines cross, they create these two angles.

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These two angles are always congruent.

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Here’s why

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EXPLANATION

First we need to remember that all lines have 180 degrees, so when we have another line intersecting the first line, it’s essentially splitting the first line into two angles, like shown:

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So, if we look at our first line (blue) we can see that the second line (red) created an angle of 145 degrees.

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Since we know that a line is equal to 180 degrees, then we know that the left over angle must be 35 degrees.

However, we also know that the 145 degree angle is also on the red line. �

So that must mean the other angle on the red line is 35 degrees as well.

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But, again, if the 35 degree angle is now on the opposite side of the blue line, then the angle next to it must be 145 degrees.

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And there we have it.

SO HOW DOES THIS HELP US?

Well, it helps because now we know that whenever we have two angles formed by intersecting lines,

The vertical angles will always be congruent.

So, for example let’s say we have something like:

We know that vertical angles are congruent

And we know that

And:

Are vertical angles, so we can say that:

And so:

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