ALTERNATE EXTERIOR AND ALTERNATE INTERIOR ANGLES
OBJECTIVE
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:
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).
LINEAR PAIR THEOREM
Statement | Reason |
| Given |
| Definition of Linear Pair |
| Definition of a Straight Angle |
| Definition of a Straight Angle |
| Angle Addition Postulate |
| Substitution |
| Definition of Supplementary Angles |
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:
VERTICAL ANGLES
So, now we know all about complementary angles, supplementary angles, and linear pairs
But what about vertical angles?
Vertical angles, by definition, are a pair of non-adjacent angles formed when two lines intersect.
So, in plain English, basically when two lines cross, they create these two angles.
These two angles are always congruent.
Here’s why
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:
So, if we look at our first line (blue) we can see that the second line (red) created an angle of 145 degrees.
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.
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.
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:
________
2 2
Now that we’ve reviewed, let’s look over what we need to go over.
SOME IMPORTANT DEFINITIONS TO GO REMEMBER
To make sure we don’t forget what we have learned, here are some definitions we need to review over just in case. This will also help us when we begin talking about the other different types of angles.
Parallel lines – Parallel lines are lines in a plane that are always the same distance apart, i.e., parallel lines are in the same plane and never intersect.
Transversal - In geometry, a transversal is a line that passes through two lines in the same place at two distinct points.
In other words, a transversal line passes through two lines, like so: (picture of parallel lines with a transversal)
Corresponding Angles
Alright, let’s start off with the definition of corresponding angles.
Corresponding angles – when two lines are crossed by a transversal, the angles in matching corners are called corresponding angles.
So, something like this: (insert the picture of the two parallel lines with a transversal and highlight the corresponding angles).
Okay, so why is this a thing?
Well, here’s why
Explanation
Okay, we know that we have parallel lines right?
Which means if the transversal is a straight line, and it crosses across the first line, it creates a certain angle.
Since the transversal is still a straight line, and the second line is parallel to the first line, the time that the transversal crosses over the second line will create the same angle.
So, this let’s us know that corresponding angles are congruent if, and only if, the transversal intersects two parallel lines.
NOW THAT WE HAVE REVIEWED, WE CAN GO OVER THE OTHER TWO ANGLES
So now that we remember what corresponding and vertical angles are, we can continue forward with alternate exterior angles and alternate interior angles.
Although, we already proved this because of how we saw different angles are always supplementary (add up to 180 degrees), it’s important to store this away so it’s less work for you later on.
So, without further ado…..
ALTERNATE INTERIOR ANGLES
So, we’ve actually already seen alternate interior angles in the homework, and you actually already proved them, but let’s take a look at them anyway.
Alternate Interior Angles - If two parallel lines are transected by a third line (a transversal), the angles which are inside the parallel lines and on alternate sides of the third line are called alternate interior angles.
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Or, again, in plain English, if you have two parallel lines intersecting with a transversal, the angles on the inside are equal.
So, I know that explanation doesn’t make that much sense, so something like this:
EXPLANATION
So, the way we know this works starts as so.
First, we need to start with two parallel lines and a transversal with one of the angles read.
Now we want to focus on the red line only.
What we can see here is that we have an angle of 145 degrees, and since the red line must add up to 180 degrees, then the angle next to it must be 35 degrees (definition of supplementary angles.
Now, let’s focus on the bottom blue line that holds our 145 degree angle.
As we can see, on the blue line, we have an angle of 145 degrees, and just like we had before, we know that all angles on the blue line must add up to 180 degrees.
So, the remaining angle must be 35 degrees as well.
But, now that we look at it, doesn’t that prove what we wanted to say?
So, this is how alternate interior Angles are congruent.
ALTERNATE EXTERIOR ANGLES
Similar to Alternate Interior Angles, we’ve already seen Alternate Exterior Angles.
So, as is tradition, let’s start with the definition.
Alternate Exterior Angles - Alternate Exterior Angles are a pair of angles on the outer side of each of those two lines but on opposite sides of the transversal.
Again, in English, this means that when you have two parallel lines intersecting with a transversal, the angles on the outside are equal.
Again, we’re looking at something like this:
EXPLANATION
The explanation as to why this is true, is very similar to alternate interior angles.
So first, like we did with alternate interior angles, let’s start off with two parallel lines intersected by a transversal.
And just like before, let’s grab the measurement of angle DBC.
So:
Now, let’s focus on the blue line this time.
Again, we know that the blue line needs to add up to 180 degrees, and since angle DBC is 145, then angle DBG must be 35 degrees.
Now, let’s focus on the red line.
Taking a look at angle EAB, we can see that it’s 145 degrees.
However, again, since all angles on the red line must add to 180 degrees, then we can safely say that angle EAH is 35 degrees.
And there it is! We have just shown that alternate exterior angles are congruent!