Properties of Special Parallelograms – rhombus, rectangle, square 128-2.24

Properties of Special Parallelograms – rhombus, rectangle, square 128-2.24


This video is provided as supplementary
material for courses taught at Howard Community
College and in this video I’m going to talk about the properties
special parallelograms. Specifically that will be the rhombus, the
rectangle, and the square. So let’s start with the
rhombus. The formal definition for a rhombus is that it’s a parallelogram with two
adjacent sides congruent. We also have two
properties for a rhombus. All sides are congruent and the
diagonals are perpendicular. So let’s start with that definition: a
parallelogram with two adjacent sides congruent.
Here’s what means. If I’ve got parallelogram ABCD, then two of the adjacent sides, let’s say side AB and side BC, will be
congruent, they’ll be the same lengths. Now we also know that in any parallelogram, the opposite
sides are congruent. So that would mean that since this is a
parallelogram as well as a specific type, a rhombus,
side AB is congruent with side CD, and side AD is congruent with it’s
opposite side, side BC. So that means all the sides
are congruent. Very often, when people talk about a rhombus, rather than use the formal definition,
they just say it’s a parallelogram with four equal
sides, and that’s what we’ve got here. The second property the rhombus has that isn’t true of all parallelograms is that
its diagonals are perpendicular. So let’s look at that.
I’ll draw in the two diagonals. So I’ve got BD and line AC. And at the place where they Intersect,
I’ll call that point E. The second property says that these two diagonals are perpendicular to each other, which
would mean the form four right angles where they Intersect. Let’s check that.
If I look at triangle ABE and triangle CBE, I can show that those are congruent
triangles. Side AB is congruent with side BC, they both share side BD and side AE is congruent with site EC because AC, that whole line, is a diagonal and for any parallelogram the diagonals bisect each other. So if AC has been bisected, that means that
AE is congruent with EC. So now I’ve got two congruent triangles.
The angles that I’ve got here, at E, are corresponding
angles for those two triangles, so they must be the
same. But they’re supplementary angles. If I’ve got two equal supplementary
angles, they must be 90 degrees each, so that
they can add up to 180 degrees. In the same way, I could take the other two
angles at that intersection and show that they are also
90-degree, or right angles. So what that means is that the diagonals are perpendicular, where they bisect they form right angles. So those are
the qualities that a rhombus has that aren’t necessarily true of all
parallelograms – all the sides are congruent and the diagonals are perpendicular to each other. Let’s go
on to the rectangle. Our definition of a rectangle is that it’s a parallelogram
with a right angle. And then we have two properties: all the
angles are right angles, and the diagonals are congruent. So the definition once again says that it’s a
parallelogram with a right angle. So if I take this
rectangle ABCD and I label one of the angles as a right angle, I can very easily show
that all the angles are right angles. I know that in a parallelogram the consecutive angles are supplementary. Well, if I have two angles like A and B and they’re supplementary, and one
angle is 90 degrees, than the other one must be
90 degrees so that they can add up, those two 90 degree angles, to
equal 180 degrees. And then I can go around the rest
of the rectangle and mark all four of the angles as being right angles. So the property all the angles are right
angles is going to be true of any rectangle, and
a lot of times when people talk about rectangles, they say it’s a parallelogram with four right angles. The second property says that the diagonals
are congruent. Let’s draw the diagonal in. So I’ll draw line BD and line AC, and I want to show that those two
diagonals are the same length, that they are
congruent. So let’s look at two triangles that they form. I’ve got triangle BAD and triangle CDA. if I can show that those two triangles
are congruent, then I can show that those long sides for each triangle, BD and CA, are congruent. Well we know that the opposite sides of any parallelogram are congruent. So BA must be congruent with CD. we know that angle A is right angle and angle D is a right angle. So we’ve got two sides that are congruent, two angles that are congruent, and both the triangles share side AD. So we’ve got side-angle-side congruency. That’s going to mean that triangle BAD is congruent with triangle CDA, and that AC must be congruent with BD. So that would be your proof that the diagonals are congruent. So once again, for a rectangle the two
specific properties they have that are not true of all parallelograms is all
the angles are right angles and the diagonals are congruent. And the last one we have is the square. A square is a rectangle with two
adjacent sides congruent. And the properties for it say that all
angles all right angles, all sides are congruent, and the
diagonals are perpendicular. Okay, so if all the angles a right angles…
Well, that make sense because the square is a
rectangle, and in a rectangle all the angles
are right angles. So a square is just a special form of a
rectangle. All sides are congruent. Well we know that two adjacent sides are congruent, which basically is the definition we’re
going to have for a rhombus. So a square is also a
special kind of a rhombus, which means all four sides are congruent. And the diagonals a perpendicular.
Once again, a square is a rectangle, and since a rectangle has diagonals that are perpendicular, the diagonals of a square will also be
perpendicular. Okay, so those are the properties for the
rhombus, the rectangle and square. Take care, I’ll see you next time.

16 thoughts on “Properties of Special Parallelograms – rhombus, rectangle, square 128-2.24

  1. At 8:12, I think you meant to say a square is a rhombus; therefore, the diagonals must be perpendicular because any rhombus' diagonals are perpendicular. Rectangles' diagonals are not necessarily perpendicular, but they are congruent.

  2. Lol community college, I'm in grade 8 and we learned this in 20 minutes you don't even need to explain this far if you have any common sense you can eyeball it lol and use a limited amount of terminology to further explain pffffff

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