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 quad, quadrilateral like these to have four completely different side lengths and four completely angles.

有时测试会询问不规则的四边形。测试更多，更有可能询问非常对称的四边形，以及 我们将讨论此视频中的那些。所以当然那些是不规则的四边形。 该组四边形还包含一些精英对称的成员，梯形，平行四边形，矩形，菱形和 the most elite of all the square.

这应该得到评论。当我们到达广场时，我们会再次谈论这一点。 但是，关于一个正方形的有趣的事情是，如果你想到它，那是你在一个小孩子时学习的第一个形状之一。 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. 当我们达到它时，我们会谈谈这个。首先，让我们谈谈什么是真的 所有四边形，绝对是成员的集合。对于每个四边形，四个内部角度的总和为360度。

你需要知道。理解这一点的一种方法是看到每个四边形， 可以分为两个三角形。所以在这里，我们有一个随机的四边形，我们将b到d从b绘制。 And we can see, we have two triangles. In triangle ABD, we have the three blue angles.

他们必须在三角形BCD中加入180。我们有三个红色角度。 Those have to add up to 180. And really, the angles in the whole quadrilateral, ABCD. 这只是红色角度的总和，加上蓝色角度。所以，红色加蓝色必须等于180加180，即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, 被称为对角线。三角形没有对角线，但每个四边形都有两个对角线。 所以，这是一个trang，这是一个四边形，其两个对角线绘制。

段和FG是四边形EFGH的对角线。正如我们将看到的，某些四边形具有具有特殊属性的对角线。 现在我们可以开始谈论特殊的四边形，在测试中更常见的更精英四边形。 The Parallelogram. 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. 所以AB等于CD，BC等于广告。属性3，相反的角度相等。

所以红色角度相等，蓝色角度相等。和财产第4号，对角线互相分解，所以 their point of insert, intersection M is actually the midpoint of each diagonal. And so we could say that AM equals MC, and separately, BM equals MD. 因此，这四个属性非常重要。我会将那些称为“四个四”平行图属性。

And here's the interesting thing. They always come together. 他们总是作为包裹交易。也就是说，如果其中任何一个是真的， 它全部意味着其他三个必须是真的。如果其中任何一个都不是真， it automatically makes the other three not true.

因此，构建一个大四四个等四边形是绝对不可能的。 Either a quadrilateral has to have all four of them, or has none of the four of them. 这就是为什么他们这么重要。并且任何四边形都是这四个真实的，是平行四边形。

Again these four properties are parallel opposites sides, equal opposites sides, equal opposites angles, and diagonals bisect each other. 所以那些人自动让其他三个真实。现在我们可以谈论菱形。 Rhombuses are equilateral quadrilateral. That is a quadrilateral with four equal sides.

Some people think about this as a diamond shape especially if we orient it this way. If we orient it with the four points pointing horizontally and vertically, 钻石只是一种，a，休闲或口语方式来引用菱形。所以菱形是平行四边形的。 所以他们自动拥有“大四”属性。每个菱形都有“大四四”属性，我们刚刚谈过。

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.

对角线属性是可分离的。来自另一个。 所以你可以有一个不规则的四边形，没有大四，没有相同的边，但它确实具有垂直的对角线。 单独的属性可以与其他四个分开。它不像四个属性，总是走到一起。

矩形。 Rectangles are quadrilaterals with four 90 degree angles. We could call them equiangular quadrilaterals. It's very interesting, with a triangle, the only equal, 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 two any higher polygons, that you can have the eque angular shape without the equilateral shape. 所以矩形具有所有相等的角度。事实上，其中一个矩形，EFGH，是一个金色的矩形。

Rectangles are parallelograms, and the "big four" parallelogram properties are true for them. 此外，还有两个特殊的rang，矩形属性。显然，所有四个角度彼此相等。 And the diagonals are congruent. So QS equals 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, that property can be separated out from the other four. 重要的是要欣赏这一点。最后，其中，这套，我们将谈论广场。

正方形是最精英四边形，形状具有最多的特殊属性。 一个方块是一个矩形。一个方形是菱形。 And a square is a parallelogram. So, it has all of the rectangle properties.

All the parallelogram properties, all the rhombus properties. And so it's a very, very special shape. If we're told that a figure is a square, now that's amazing. The test problem actually says this shape is a square, they're giving us a ton of information. And that's, that is a really powerful thing to know, there's all kinds of 几何事实我们知道我们是否只需具有形状是正方形的信息。

但是，难以证明某事是广场。假设形状是正方形的，不要轻容 当您没有足够的信息来执行此操作。这是测试上的一个非常常见的陷阱。 If the shape is close to being a square, but not exactly a square, it doesn't necessarily have ANY of the square properties.

So here are two drawn-to-scale diagrams. Both of these look like squares, but neither is. 所以左边的一个，EFGH，结果是菱形。但它有一个角度到四面是相等的，但是 one angle is slightly less than 90 degrees. The other angle is slightly more than 90 degrees.

So it's not exactly a square. The other one has three equal sides, and 然后有一边有点少。k，kl有点少。 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. So it's just the fact that even if we have something that is drawn to scale and looks like a square, there is no guarantee that it is a square. Here's a practice problem. 暂停视频，然后我们会谈谈这个。 好的，这是一个非常奇怪的问题格式。

Can we determine that ABCD is a square, if we know either of these? So BC equals CD, and angle B is 90 degrees. So that's fact number one. AD equals AB and angle D equals 90 degrees. 所以问题是，只使用其中一个，我们可以确定它是一个广场吗？如果我们把两个人放在一起，那就足以确定它是一个广场吗？

或者，如果我们把它们俩放在一起，那么证明某事是正方形的不够。 事实证明，如果两个事实都是真的，那就不保证形状是正方形。 它可能只是两个一致的右翼三角形，如此。

所以在这个图中，BC等于CD是真的。ad等于ab是真实的。 我们确实有正确的角度和B和D，但ABCD不是正方形。事实上，它根本不是任何特殊的四边形。 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 sides are called "legs".

腿上的两个角度总是补充。所以其中一个是90度，另一个必须是90度。 例如，它总是如此，例如A Plus B等于180，C加D等于180.而且是因为平行线的基本属性。 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 equals LM the legs are equal, 然后我们知道相对侧的角度必须是相等的。基本上，形状完全是对称的。 So, angle K equals angle L. Angle J equals angle M.

And also the diagonals have equal length. 这是一个练习问题。暂停视频，然后我们会谈谈这个。 Okay, ABCD is a trapezoid with lengths shown.

Find the diagonal AC. Well there's no direct formula. 我们无法插入我们拥有的四个数字，并找到对角线。 We're gonna have to find this step by step. Essentially we're gonna be working our way up to the pythagorean theorem.

作为一般规则，在任何几何问题中，您被要求找到倾斜线的长度，机会非常， very good that the pythagorean theorem is hidden somewhere in that problem. And your job is just to 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 draw perpendicular segments from BC down to the base.

So what we create, we have some kind of rectangle in the middle. Looks like might be close to a square, but its not exactly a square. 然后，我们每侧有两个对称的右三角形。所以我们知道BC，EF的另一侧。 这也必须是11.整个基础是21岁，我们知道EF， AE and FD those, 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. 因此，我们将基座5,11,5的区域分开，并且增加到21个区域。 Well now. Notice in those right triangles, we have 5 blank 13.

应该响铃。这是一个5-12-13三角形。 所以必须是真实的，也必须等于12。 So now we know the length of the height. Now we can think about that, that diagonal.

该对角线AC，是右三角形ACF的斜边。我们知道AF是5加11,16和CF是12。 Well, this is the 3, 4, 5 triangle scaled up, scaled up by a factor of 4. It's a 12, 16, 20 triangle. 所以斜边AC是20.这是对角线的长度。

所以，请注意，我们发现了一切。使用毕达哥拉斯定理。 And 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. This diagram shows the conceptual relationship among the quadrilaterals.

So, first of all, there are many quadrilaterals that are neither parallel, parallelograms or trapezoids, so 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. Secondly, 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 subtle, more symmetrical, and special kinds of quadrilaterals, 因为那些有更多的属性，所以还有更多的询问。这就是为什么测试喜欢它们。 总之,这是适用于所有的四边形the sum of the angles is 360 degrees.

知道大四平行四边形属性，平行的相对侧，相等的相对侧​​，相反的角度，以及相反的角度非常重要 diagonals bisect each other. Those four always come together. So you can't have, any shape that has some of those true, and others of them not true.

They're either all four of them true, or all four of them false about a particular shape. 菱形有四个相等的方面加上“大四”。它也具有垂直的对角线。 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, 和所有平行四边形属性。梯形是，具有一对平行侧，和 对称梯形或等腰梯具有相等的长度。这意味着它在每侧具有相同的角度，以及相同的对角线。