Now we can start talking about geometry. And geometry, of course, is the study of shapes. 现在，对于某些视觉导向的人，几何是自然的。在其他没有开发他们的视觉技能的人中， geometry can be a little bit harder. So especially for the folks for whom geometry is a little bit harder, here's what I'm gonna say.

It is not enough simply to watch these videos. After you watch these, get out paper and a ruler and draw these different shapes, actually physically draw them on paper. And build shapes from physical objects. You can use pencils, toothpicks, straws, anything like that. Actually build triangles, build rectangles, actually look at them.

Use your hands. Our hands are actually part of our intelligence. So if you are using your hands you're engaging every part of the brain, and it will make it much easier to understand all these relationships. So let's start with lines. Lines are straight and they go on forever in both directions.

所以在这里，我们在一堆不同的方向上有一束不同的直线。 你必须想象的每一行there are some arrows or something like that to indicate that the lines 实际上确实在两个方向上都会继续。非常重要的是不要用水平混淆。

Those two words have very different meanings, but sometimes there are some students who confuse them. 所有线路都是直的。所以我们在上一张幻灯片上的所有线条， 线路进入不同的方向，所有这些都是直线。你可以随时假设一条线直接在测试中。

If it looks straight, it is straight. That is always true on the test. 但是，一些线条是为了方便地水平绘制，但你永远不会假设线路完全是水平或垂直的，因为它们出现了。 Now people get really confused on this. If you're confused, if you think that horizontal and straight mean the same thing, then when we say you can assume from the test that lines are straight, people mistakenly assume that this also means they can assume lines are horizontal, and that is not correct.

A line segment is a finite piece of a line. So for example, here we have a line segment. It has two endpoints, and when these endpoints are labeled, that makes it easy to discuss. This is line segment AB. And for the purposes of the test, AB can either mean the actual shape, the line segment itself, or it can mean the length of the line segment, the numerical length.

An angle occurs between two lines or two segments. So for example, here we have an angle. 这恰好在一行和一个段之间。理解角度的最佳方式是动态地想到它， as the act of turning or rotating. So in other words, going from here to here.

That is what an angle is. It's that dynamic space in between the two lines. 如果我们标记点，我们可以谈论一定角度。我们可以称之为CDE或EDC角度。 点D，角度的顶点，右边，角度的点，必须在名称中间。

And so we can call it either CDE or EDC, as long as the vertex is in the middle. Sometimes in these videos, I'll also use a single angle name if there's no ambiguity. 例如，在该图中只有一个角度，所以我可以理论上调用它角度D.这可能发生在测试中，尽管测试是 often careful enough to use a three-letter name always for an angle. We measure the size of an angle in degrees.

The test can state these directly, so 50 degrees. Alternately, the test can label the diagram and state the measure of the angle in the text. So angle GFH = 50 degrees. Because they put letters on the points in the diagram, we can just use that to talk about the measure and the number of degrees in the text.

Actually, probably its favorite thing to do is the following. Just specify an angle with a variable number of degrees. This flexible format allows them either to specify the angle, for in the text they could say x = 50. 或者他们可能会提出一个关于它的问题。他们可以给我们其他信息并说出x。

So they like doing this. We'll do a quick review of basic degree facts. In a straight angle there are 180 degrees. But of course, remember a straight line can go in any direction, but if there's any point on the straight line all the way around from one side of the line to the other, that's 180 degrees.

直角有90度。所以在这里，我们有两条线以直角交叉。 There are actually four right angles at that intersection. If the two lines or segments meet at right angles, they are called perpendicular. 这是一个你应该知道的术语。测试可以绘制那个小方块，垂直标志 which is that little square, or it can indicate that the angle's 90 degrees.

它可以在图中标记90度或具有x度并告诉我们x等于90的文本。 所以有多种方式可以告诉我们它是90度的角度。如果不是，请不要假设两条线是垂直的 明确地被告知。这通常是陷阱。

Suppose these points appear as part of a larger diagram and no further information is given. 当然看起来那些可能是正确的角度，这是一个非常诱人的人。 测试似乎是为了假设这些线垂直的错误，并且角度恰好等于90度。

事实上，它没有。我绘制了这一点，那个角度有89.6度的角度。 So it's close to being a right angle, and it may look like a right angle to the naked eye, but none of the special right angle properties are true. And in upcoming videos, we'll be talking more about special right angle properties. None of the special right angle properties are true if the angle is close to 90 but not exactly 90.

很重要。所以你不能认为两条线垂直 除非你有某种理由这样做。一个术语我将介绍，可能不会出现在测试中 congruent. Congruent is like equal for shapes.

We use the concept of equal for a number and the very similar concept of congruent for shapes. Two shapes are congruent if they have the same shape and the same size. They don't have to have the same orientation. So for example, the purple and the green shapes here are congruent. One is flipped over from the other.

One, you could say, is a right-handed version and the other is a left-handed version, but it is the same shape fundamentally. These two are congruent, even though they have different orientations. A bisector cuts something into two congruent pieces. An angle bisector cuts an angle into two smaller congruent angles. So for example, here we have an angle bisector.

If we're told, for example, that the big angle PNM is 40 degrees and that NQ bisects the angle, 然后我们可以推断出两个较小的角度必须是20度。他们每个人都必须恰好一半，彼此相等， because the angle was bisected. Similarly, the bisector of a segment may be a point, another segment or a line.

The bisector divides the segment into two equal halves. So notice here segment ST bisects PQ. Also notice it's definitely true that PQ does not bisect ST, because SR is clearly bigger than RT. 因此，ST双分解PQ意味着R是PQ的中点，并且该PR等于RQ。

我们已经将其分为两半。而且，这总是一直是什么比较的手段。 Sometimes a line will bisect a segment and also be perpendicular to it. The line is called a perpendicular bisector of the segment. Line VW here is the perpendicular bisector of TU. Every point on the perpendicular bisector of a segment is equidistant from the two endpoints on the segment.

所以这是一个真正y handy fact to know. That shows up in a variety of ways. The perpendicular bisector, in fact, is the set of all possible points that are equidistant from the two endpoints of the segment. Now some basic facts about angles. We've already said that a straight line contains 180 degrees.

这意味着，如果两个或更多个角度位于直线上，则其角度的总和是180度。 例如，我们可以假设长线是直的。这一点没有某种轻微的弯曲。 The test will not do that to us. If it looks straight, it is straight.

And therefore, we know that those two angles together make 180. So x + y = 180. 如果两个角度最多增加180，则它们称为补充。直线上的两个角度始终补充，所以，P + Q = 180。 When two lines cross, four angles are formed. So here we have two lines.

They're going on forever in both directions. They happen to cross, and these four angles are formed. The pairs of angles opposite each other, sharing only the vertex in common, are called vertical angles. 垂直角度总是一致。例如，a和c。

They don't share any sides. All a and c have in common is they touch at a single vertex. 他们在顶点触摸。B和D还在顶点触摸。 And so that's why they're called vertical angles, cuz they meet at a vertex. So we know the vertical angles are congruent.

We know that a = c and b = d. Of course, the pairs of angles next to each other, a + b, b + c, 所有这些都是补充的。它们都加到了最多180度，因为我们在一条线上有成对的角度。 因此，如果我们在该图上给出了一个角度，我们可以找到其他三个角度。例如，如果a = 35，我们知道c必须相等， 这也必须是35度。

And b and d have to be the supplementary angle of 145 degrees. So that any two pairs together, any two angles together in a pair, 高达180度。这是一个练习问题。 Pause the video, and then we'll talk about this. Okay, in the diagram x = 40 degrees and RT二等于大角度，SRU，这是一个非常大的角度。

Well, SRU is the supplementary angle to that 40-degree angle, so SRU has to be 180 minus 40 which would be 140. So SRU is 140. And this angle is bisected. Because it's bisected, it's cut into two equal halves. So those two halves, each one has to be 70 degrees.

srt = 70度，tru = 70度。那些是二等的角度的二等分半。 Well, now notice that the angle TRV, that angle is made of TRU and angle x, which we know. We know TRU is 70 degrees. We know that angle x is 40 degrees.

So we add them together. TRV has to be an angle of 110 degrees. Now notice that TRV is the vertical angle of SRW, so those two have to be equal. So that means that SRW also has to be a 110-degree angle, so y = 110. 最后，我们将审核平行线。如果两条线是平行的，它们从未相交和 they're always exactly the same distance apart.

And again, this is another one of these properties like perpendicular, close to parallel doesn't count for beans. 你必须知道这两条线都完全平行。显然，由于平行线从未相交， they never form angles with each other. We get many angles, though, if a third non-parallel line cuts across the two parallel lines.

This third line is called a transversal. A transversal is a line that cuts across two parallel lines. So here we have a transversal cutting across the parallel lines WX and YZ, and we get eight angles there. Now the four big angles are all equal, and the four little angles are all equal. So in other words, a = d = e = h，b = c = f = g。

这是一个很大的想法。现在，当然，你可能会记得几何形状， 有各种特殊名字。交替的内部和相同的侧面和相应的角度。 If you want to remember all those special names, that's great. You don't need to.

All you need to remember is all the big angles are equal, all the little angles are equal. So here's the diagram again, and now I've labeled it so that it's clear that everything is equal. 另请注意，P和Q是补充的。所以任何大角度加上任何小角度都等于180度。

这是一个非常重要的想法。因此，如果我们在这里被赋予任何一个角度的程度， 我们可以找到其他七个。总之，我们谈到了线条和线段。 我们谈到了角度和程度。我们指出，直角有180度 90 degrees in a right angle.

We talked about angle bisectors and perpendicular bisectors. An angle bisector divides an angle into two smaller equal angles. A perpendicular bisector is perpendicular to a segment and divides it into two equal halves. We talked about how two angles on a line are supplementary. Vertical angles are congruent.

And we talked about the angles formed by a transversal intersecting a pair of parallel lines. 我们将在即将到来的视频中讨论这些基本思想的许多应用。