Hello everybody, how are you doing? This is Mr. Dallas, and in today's lesson I'm going to discuss properties of parallelograms. But before I do that, I think I need to go over the definition of a parallelogram. A parallelogram is a quadrilateral where opposite sides are parallel. So a parallelogram is a four-sided polygon where opposite sides are parallel.
So I've got an example of a parallelogram up here, and I am marking each set of opposite sides as being parallel to each other. So the arrows here indicate that these are parallel to each other. So this is a parallelogram right here.
And if you have a hard time remembering the definition of a parallelogram, the word parallelogram has parallel in it. And so when I was a student in high school, that's the way that I remembered what a parallelogram was. It was sides were parallel to each other.
Again, because the word parallelogram has the word parallel in it. So let's see here. We have seven properties I'm going to discuss. that every parallelogram has.
I don't care how big the parallelogram is, how small it is, what country you're in, what language you speak. Every single parallelogram has all seven of these characteristics or properties. First one here is the opposite sides are parallel. This is the actual definition of a parallelogram.
It is a characteristic of a parallelogram. Then we have opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary. Diagonals bisect each other. One pair of sides are congruent and parallel. And then each diagonal divides the quadrilateral into two congruent triangles.
So if you count them up, there are seven properties or characteristics of parallelograms. So I'm going to go over every single one of these in a little bit more detail, because you might not understand what certain words are in here. Let's see here. The first one I have listed are that opposite sides are parallel.
Again, this is the very first thing I covered when I did the definition. Our opposite sides are parallel to each other. So I'm not going to spend much time on this one. The second one says that opposite sides are congruent.
Congruent means equal or equal to. So A would be congruent to side C in this case. And side AD would be congruent to side BC in this case.
And so let's say I measured this with a ruler. And I got, I don't know, 5 inches. If this is 5 inches here, then this should also be 5 inches as well. And then let's say I measure this one here and I've got 3 inches. If this is 3 inches here, then this is also 3 inches.
Now I didn't measure with a ruler, so I'm just making up numbers here. But the idea here is that opposite sides are going to be congruent to each other on every single parallelogram. The third property I have here are that opposite angles are congruent.
So angle A would be equal to angle C and angle B. B would be congruent to angle D. And if I pull out a protractor and let's say I got 120 degrees for angle A, then angle C should also be 120 degrees. And if angle D is 60 degrees, then angle B should also be 60 degrees. And again, I didn't measure these with a protractor.
I'm just using this as an example. But the idea here is that opposite angles should be equal to each other on every single parallelogram. The next property I have here are the consecutive angles are such a Another way of understanding consecutive is next to. Consecutive means next to.
And then a way to understand supplementary are two angles that equal 180 degrees. So if I were to kind of reread this again, I could say that there are two angles that are next to each other that equal 180 degrees. So I'm going to use the same numbers on the previous slide. to go over this in a little bit more detail.
So the idea here is that the angles that are next to each other will always equal 180 degrees on a parallelogram. And if I add 60 and 120, I get 180. And this is going to happen all the way around the parallelogram. No matter how I add up 120 and 60, it's always going to equal 180 degrees. So this is the fourth property I'm discussing on parallelograms.
the angles right next to each other in the parallelogram will always equal 180 degrees. The next property I have are that diagonals bisect each other. First of all, a diagonal is a line segment cutting across a polygon from vertex to vertex. So from A to C, I have a diagonal in red, and then from D to B, I have a diagonal.
And bisect, another way of understanding what bisect means is bisect means to cut. cut in half. So when I say diagonals bisect each other, the diagonals are cutting each other in half.
So the diagonal in red is cutting the diagonal in blue in half, and the diagonal in blue is cutting the diagonal in red in half. So I'm going to put a fifth point here. Let's call this point E.
So I'm going to call this point E. So if this right here, A, this segment should equal this segment right here. So A and segment EC are congruent to each other.
Same thing for segment DE and EB. These diagonals are cutting each other in half. So if this was, I don't know, let's say it was 7 centimeters. If this is 7 centimeters, this would also be 7 centimeters.
And let's say this whole length here was 20 centimeters. This diagonal is cutting it in half, so half of 20 would be 10 centimeters here. and like 10 centimeters here. So the idea here are the diagonals are chopping each other in half. The whole length is 14. I'm chopping it in half, though, so 7 and 7. The whole length here is 20. I chop that in half, so I'm going to get 10 and 10. So this is the fifth property of every single parallelogram.
Another property of parallelograms are that one pair of sides are congruent and parallel. I feel like I should put the word both in here. So one pair of sides are both.
congruent and parallel. So if A is congruent to DC, then for this to be a parallelogram, the sides would also have to be parallel to each other. So they're both congruent and parallel.
Or I could have chosen this side here to be congruent to this side here, and if that's the case, for this to be a parallelogram, then these sides would also have to be parallel to each other. I cannot mix and match this. I cannot say these sides are congruent, these sides are parallel, therefore I have a parallelogram.
That is a false statement. If I were to take this side here and rotate it over here, and I ignored everything right here, I have a trapezoid where these sides are parallel to each other and these sides are equal. Is this a parallelogram? Are these sides right here parallel to each other? No, they're not.
So right here, You cannot mix and match this. You've got to have both set of sides are parallel and congruent all at one time for this to be a parallelogram. The very last property that I have is that each diagonal divides the quadrilateral into two congruent triangles. So I have two different diagonals here. I've got the diagonal in blue.
Now I've got a different diagonal in red. So this is going from D to B. This one's going from A to C. If you chop a parallelogram through the middle like this, you get two triangles that are equal to each other.
So if I were to take this triangle, let's say I highlight this triangle in green. If I were to rotate this triangle around, it would look... exactly like the triangle I have on the bottom here.
I'm doing a very bad job of drawing here. I apologize. I'm not doing any justice here. But the idea here is that this triangle in blue that I have would be equal to the triangle in green.
So these are equal to each other. If I rotated this triangle around on the top left, it would match up exactly with this triangle I have here in blue. Same thing is going to happen over here. This triangle Right in here, this triangle right here would be congruent to this triangle I have on the top right. So the idea is each diagonal divides the quadrilateral into two congruent triangles.
So these are the seven characteristics or properties of parallelograms. With this, I hope you have a better understanding of parallelograms. And remember... Every single parallelogram is going to have all seven of these properties.
Anyways, I hope you have a great day. Bye-bye.