Hey everyone, it's Justin again. In this lesson, we're going to focus on reviewing the properties,
theorems, and definitions you've learned so far in this course, and applying them as justifications
for statements. By the end of this lesson, you will be able to apply previously learned
properties, definitions, and theorems to justify statements. First you will complete the property
review activity in the PDF. Next we'll review the difference between equality and congruence.
Finally we'll complete three practice problems.
In your notes template PDF for this lesson, you'll find an activity titled property review.
you'll want to complete this activity before continuing on with this video. We're not going
to be going over the answers together, so be sure to check in with an adult and check the
answer key before resuming this video. Pause here to complete the property review activity.
If you're back, you should have already completed the property review activity in your notes and
have checked your answers. Let's move on to the next item on our agenda. Congruent and equals are
both used to show that two things are identical. So it can be tricky to remember when to use an
equal sign and when to use a congruent sign. The easiest way to remember the difference is that an
equal sign or equality only applies to numbers, while congruence applies to lines, angles, and
figures. Let's practice a little sorting. Here are six statements. Look them over and decide whether
the blank should be filled with an equal sign or a congruent sign. I'll give you a hint and tell you
that there are going to be three of each in the end. Pause the video here to sort them yourself.
Let's start with angle DOG and angle CAT. These are referring to the angles, so that means I
need congruence. Now if we look at the measure of angle DOG and the measure of angle CAT, this is
referring to how big the angles are in degrees, which is a number. So this one is equality.
Take a look at these two together next, since there's only a very small difference between them.
The one on the right has the segment notation above LA and MP, while the one on the left does
not. If you think way back to unit one, can you remember what the difference is between these?
When we use the segment notation, we are referring to the segments themselves.
When there's no segment notation, we're referring to the length of the segment.
Knowing that, how would you categorize each of these?
When we have segment notation, we need congruence. When there's no segment notation,
we would use equality since we're talking about length, which would be a number. Two more to go.
How about triangle GLD and triangle FSH? This is talking about the whole triangle, so this is
congruence. Lastly we have 5 plus 8 and 13. Those are all numbers, so we are talking about equality.
Here's our first example. There are two questions that we have here about the quadrilateral on
the left. The first asks, which two angles are alternate interior angles formed by PA and RT?
This diagram looks a little different than back when we learned about transversals in unit 3,
but we can visualize it in a way that's more familiar. Maybe something like this?
All I've done here is extend PA, RT, and RP, and I've temporarily made BT and RA a little
less visible so that they don't distract us. I have not actually changed the diagram at all.
This is a great example of how you can expand on the diagram on your
paper and help yourself visualize things in an easier way. If you couldn't tell before,
go ahead and identify which two alternate interior angles are formed by PA and RT.
Did you identify angle 2 and angle 3, since they are on the opposite sides of the transversal that
crosses PA and RT? Nice job. Can you see how angles one and four aren't actually formed by
PA and RT? The other interior angles formed by PA and RT would be these two much larger
angles that were not a part of our original diagram. Now let's look at our other segments.
Which two angles are the alternate interior angles formed by RA and PT?
Take a look at this adjusted diagram to help give you a better picture.
Angles 1 and 4 are alternate interior angles formed by RA and PT. If you ever find yourself
using a diagram like this, you can always sketch on it like I've done here to help
make it easier for you to interpret. Just make sure that you don't add in any new information.
Next up is example two. We have a new diagram with four new angles. Let's work our way down
the list of statements and questions on the right, starting from the top.
Given the diagram to the left, angle 1 and angle 3 are blank. This is a great preview of the process
of writing out proofs. You always want to start as simple as possible. So don't worry about the
relationship between their measures yet. What type of angles are one and three together?
They're vertical angles. How do you know they're vertical angles?
Your answer for this one should be one of the properties from the property review. Since we're
dealing with lines and angles here, I'll give you a hint that you'll find it in section three.
What property can be used to show how we know angles one and three are vertical angles?
We can use the vertical angles definition, because this helps us identify vertical angles from a
picture. Now that we know that angles 1 and 3 are vertical angles, what else do we know about them?
Vertical angles are always congruent. Which one of those properties from the review activity
tells me that? The vertical angles theorem. You'll find that this is a very consistent
pattern throughout this unit. Whenever you want to make a statement about the relationship between
two angles or segments, you'll first need to state a definition that lets you label them,
and then you can use a theorem or postulate that allows you to state their relationship.
Let's do this last one together. Now that we know that angles one and three are congruent, I also
know that their measures are equal. Remember, since the measure of an angle is a number of
degrees, we use an equal sign for it. We know this because of the congruence definition for angles.
You'll find yourself using the definition of congruence a lot throughout this unit to move
between a congruent statement and an equation. It's one of the most important properties, because
by allowing us to write an equation, we can also do things like solve, substitute, or manipulate.
For our last example, we're going to continue using this same diagram. But
now instead of working with angles one and three, we're working with angles one and four.
Using example two as a guide, go ahead and see how far you can get on your own in example three.
Were you able to identify that angles 1 and 4 are a linear pair? We know this because of the linear
pair definition, which helps us identify them from the diagram. Once we know that angles one
and four are a linear pair, we can also conclude that angles one and four must be supplementary.
Which property lets us conclude this? The linear pair postulate.
And then the most tricky one is once we know that angles one and four are supplementary,
then what else can we say about them? Well, what do we know about supplementary angles? The sum
of their measures is 180 degrees. So we can say that the measure of angle one plus the measure
of angle four will equal 180. Which property is it that tells us this? The supplementary definition.
Now you can accurately use congruence inequality in the appropriate situations and apply the
properties we've learned so far in this course to justify statements. See you next time.