What are quadrilaterals? In this video, we explore that question and more. Enjoy — and then be sure to watch our other Geometry videos!
Now we can talk about quadrilaterals. A shape with four line-segment sides is a quadrilateral.
So here we have four random quadrilaterals. These are actually called irregular quadrilaterals. It is possible for a quadrilateral like this to have four completely different side lengths and four completely different angles?
And sometimes the test will ask about an irregular quadrilateral. The test is more likely to ask about the very symmetrical quadrilaterals. And we’ll talk about those also in this video.
So of course, those are the irregular quadrilaterals. The set of quadrilaterals also contains some elite symmetrical members, the trapezoid, the parallelogram, the rectangle, the rhombus, and the most elite of all, the square.
And this deserves a comment. We’ll talk about this again when we get to squares. But the funny thing about a square is if you think about it, it’s one of the first shapes you learn when you’re a little kid. So it’s a very familiar shape. And the fact that it’s so familiar makes it hard to appreciate how special and how elite a shape it is.
It’s very hard to prove something is a square. A square is an incredibly elite shape. And we’ll talk about this more when we get to it.
What Is True for All Quadrilaterals
First of all, let’s talk about what’s true for all quadrilaterals, absolutely every member of this set. For every quadrilateral, the sum of the four interior angles is 360 degrees, that you need to know.
One way to understand this is to see that every quadrilateral can be divided into two triangles. So here we have a random quadrilateral. And we draw the line from B to D. And we can see we have two triangles. In triangle ABD, we have the three blue angles, they have to add up to 180.
In triangle BCD, we have the three red angles, those have to add up to 180. And really, the angles in the whole quadrilateral ABCD, that’s just the sum of the red angles plus the blue angles. So red plus blue has to equal 180, that’s 360. So that’s why every quadrilateral has a sum of angles of 360. Now this line that we drew from one vertex to the opposite vertex is called a diagonal.
Triangles don’t have diagonals, but every quadrilateral has exactly two diagonals. So here’s a quadrilateral with its two diagonals drawn. Segments EG and FH are the diagonals of quadrilateral EFGH. As we will see, some quadrilaterals have diagonals with special properties. Now we can start talking about the special quadrilaterals, the more elite quadrilaterals that are more common on the test, the parallelogram.
The Big Four Parallelogram Properties
All parallelograms have the following four properties:
- Property number 1: opposite sides are parallel. This is kind of the definition of a parallelogram. So AB is parallel to CD and AD is parallel to BC.
- Property number 2: opposite sides are equal. So AB equals CD, and BC equals AD.
- Property number 3: the opposite angles are equal. So the red angles are equal and the blue angles are equal.
- And property number 4: the diagonals bisect each other. So their point of intersection M is actually the midpoint of each diagonal. And so we could say that AM = MC and separately, BM = MD.
So those four properties are really important.
I will refer to those as the big four parallelogram properties. And here’s the interesting thing, they always come together. They always come as a package deal, that is, if any one of them is true, it automatically means that the other three have to be true. And if any one of them is not true, it automatically makes the other three not true.
So it is absolutely impossible to construct a quadrilateral that has some of the big four but not others. Either a quadrilateral has to have all four of them or has none of the four of them. And that’s why they’re so important. And any quadrilateral that has all four of these true is a parallelogram.
Again, these four properties are parallel opposite sides, equal opposite sides, equal opposite angles, and diagonals bisect each other. So any one of those automatically makes the other three true.
Now we can talk about rhombuses. Rhombuses are equilateral quadrilaterals, that is, a quadrilateral with four equal sides.
Some people think about this as a diamond shape especially if we orient it this way. We orient it with the four points pointing horizontally and vertically. Diamond is just kind of a casual or colloquial way to refer to a rhombus. So rhombuses are parallelograms, so they automatically have the big four properties. Every rhombus has the big four properties true that we just talked about.
Two Special Rhombus Properties
In addition, there are two special rhombus properties. All four sides are equal, and the diagonals are perpendicular. So if you have a parallelogram with perpendicular diagonals, it has to be a rhombus. I will point out, though, it is possible to have an irregular quadrilateral that has perpendicular diagonals.
That diagonal property is separable from the others. So you could have an irregular quadrilateral that doesn’t have the big four, doesn’t have the equal sides, but it does have perpendicular diagonals. That property alone can be separated from the other four. It’s not like the big four properties that always come together. Rectangles, rectangles are quadrilaterals with four 90-degree angles.
We could call them equiangular quadrilaterals. It’s very interesting with a triangle, the only equilateral triangle is equiangular, and the only equiangular triangle is equilateral. Those two always have to come together with triangles. But we can separate those two once we get two quadrilaterals, or to any higher polygons, that you can have the equiangular shape without the equilateral shape.
So rectangles have all equal angles. And in a fact, one of those rectangles, EFGH, is a golden rectangle. Rectangles are parallelograms, and the big four parallelogram properties are true for them. In addition, there are two special rectangle properties. Obviously, all four angles are equal to each other and the diagonals are congruent, so QS = PR.
And again, this diagonal property, this can be separated out from the others. We could have an irregular quadrilateral that doesn’t have any of the big four, doesn’t have right angles, but it does have congruent diagonals. So that property can be separated out from the other four. It’s important to appreciate that. Finally, among this set, we’ll talk about squares.
What’s So Special About Squares?
Squares are the most elite quadrilaterals, the shape with the highest number of special properties. A square is a rectangle, a square is a rhombus, and a square is a parallelogram. So it has all the rectangle properties, all the parallelogram properties, all the rhombus properties. And so it’s a very, very special shape.
If we are told that a figure is a square, now that’s amazing. If the test problem actually says this shape is a square, they’re giving us a ton of information. And that is a really powerful thing to know. There’s all kinds of geometry facts we know if we simply have the information that a shape is a square.
Don’t Be Fooled!
But it’s very hard to prove that something is a square. Don’t be gullible in assuming that a shape is a square when you don’t have sufficient information to do so. That is one very common trap on the test. If the shape is close to being a square but not exactly a square, it doesn’t necessarily have any of the square properties.
Here are two drawn to scale diagrams. Both of these look like squares, but neither is. So the one on the left, EFGH, turns out to be a rhombus. The four sides are equal but one angle is slightly less than 90 degrees, the other angle will be slightly more than 90 degrees. So it’s not exactly a square.
The other one has three equal sides and then has one side that’s a little bit less, KL is a little bit less. It looks like angle M is 90 degrees, but angle K is greater than 90 degrees, and the other two are slightly less and unequal to each other. So that’s a totally irregular quadrilateral. But, drawn to scale, it looks like a square.
Just the fact that even if we have something that is drawn to scale and looks like a square, there’s no guarantee that it is a square.
Here’s a practice problem. Pause the video and then we’ll talk about this.
Okay, this is a very odd question format. Can we determine that ABCD is a square if we know either of these?
So, BC = CD, and angle B is 90 degrees. So that’s fact number 1. AD = AB and angle D = 90 degrees. And so the question is using just one of them, can we determine that it’s a square? If we put both of them together, is that enough to determine that it’s a square? Or if, even if we put them both together, it’s not enough to prove that something is a square.
Turns out that if even both facts together are true, that does not guarantee that the shape is a square. It could be just two congruent right triangles attached at the hypotenuse, like this. So in this diagram, it is true that BC = CD, it is true that AD = AB, and we do have right angles at B and D, and yet ABCD is not a square.
In fact, it’s not any special quadrilateral at all. All this information is not enough to determine that ABCD is a square and the answer to the question is C.
Now we can talk about trapezoids. A trapezoid has exactly one pair of parallel sides. So these are trapezoids.
It is possible for a trapezoid to have two right angles in it on one of the legs. The two parallel sides are called bases and the non-parallel are called legs. The two angles on a leg are always supplementary. So one of them is 90 degrees, the other has to be 90 degrees. It’s always true that for example, A + B = 180 and C + D = 180. And that’s because of the basic properties of parallel lines.
Some trapezoids have two equal legs. We call these symmetrical trapezoids, or sometimes a more formal name for them is isosceles trapezoids, either term is fine. If the two sides are parallel, and if KJ = LM, the legs are equal, then we know that the angles on the opposite sides have to be equal. Essentially, the shape becomes entirely symmetrical.
So angle K = angle L, angle J = angle M, and also the diagonals have equal length.
Here’s a practice problem. Pause the video and then we’ll talk about this.
Okay, ABCD is a trapezoid with length shown. Find the diagonal AC. Well, there’s no direct formula.
There’s no way we could just plug in the four numbers we have and find the diagonal. We’re gonna have to find this step by step. Essentially, we’re gonna be working our way up to the Pythagorean theorem. As a general rule in any geometry problem where you’re asked to find the length of a slanted line, chances are very, very good that the Pythagorean theorem is hidden somewhere in that problem.
And your job is to just figure out how to use the Pythagorean theorem. That’s a really big idea. So here what we’re gonna do is we’re gonna drop perpendicular segments from B, C down to the base. So what we create, we have some kind of rectangle in the middle looks like it might be close to a square but it’s not exactly a square.
And then we have two symmetrical right triangles on each side. So we know that the opposite side from BC, EF, that also has to be 11. Well that whole base is 21. And we know that AE and FD, the two small sides of the triangle, those have to be equal to each other, cuz those triangles are congruent. So it must be true that each one has a length of 5.
So we split up the area of the base, 5, 11, 5, and that adds up to 21. Well now, notice in those right triangles, we have 5 blank 13. That should ring a bell. This is a 5-12-13 triangle, so it must be true that BE and CF = 12. So now we know the length of the height, Now we can think about that diagonal. That diagonal AC is the hypotenuse of right triangle ACF.
And we know that AF is 5 plus 11, 16, and CF is 12. Well this is the 3, 4, 5 triangle scaled up by a factor of 4. It’s a 12,16, 20 triangle. So the hypotenuse AC is 20, that’s the length of the diagonal. So again, notice that we found everything using the Pythagorean theorem. Once again, whenever you have to find the length of a diagonal or really the length of almost any slanted line, chances are very, very good that it’s a Pythagorean theorem problem.
Conceptual Relationship Among Quadrilaterals
This diagram shows the conceptual relationship among the quadrilaterals. First of all, there are many quadrilaterals that are neither parallelograms or trapezoids. Those are just the irregular quadrilaterals that are outside the two big circles. Inside the parallelogram circle, everything in that circle has the big four parallelogram properties.
And nothing outside the parallelogram circle can have any of those properties. Inside the parallelogram circle, we have rhombuses, rectangles, and then squares are the intersection of rhombuses and rectangles. Because squares are both rectangles and rhombuses. And of course, they’re parallelograms also. Separately, we have the trapezoids.
Within the trapezoids, we have the region of symmetrical trapezoids, a special case. And again, the test most likes to ask about the more elite, more civil, more symmetrical and special kinds of quadrilaterals. Because those have more properties. And so there’s more to ask about, that’s why the test likes them. In summary, it’s true for all quadrilaterals if the sum of the angles is 360 degrees.
It’s very important to know the big four parallelogram properties, parallel opposite sides, equal opposite sides, equal opposite angles, and diagonals bisect each other. Those four always come together. So you can’t have any shape that has some of those true and others that are not true.
They’re either all four of them true or all four of them false about a particular shape. A rhombus has four equal sides, plus the big four, it also has perpendicular diagonals. Rectangle has 90-degree angles plus the big four. It also has congruent diagonals.
A square is a rectangle and a rhombus. So a square has all the rectangle properties, all the rhombus properties, and all the parallelogram properties. A trapezoid has exactly one pair of parallel sides. And a symmetrical trapezoid or an isosceles trapezoid has equals lengths. This means it has equal angles on each side as well as equal diagonals.