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CA Geometry: More proofs

7-10, more proofs (10 continued in next video). Created by Sal Khan.
Video transcript
All right, we're on problem number seven. And when I copied and pasted it I made it a little bit smaller. So I'm going to read it for you just in case this is too small for you to read. It says, use the proof to answer the question below. So they gave us that angle 2 is congruent to angle 3. So the measure of angle 2 is equal to the measure of angle 3. I'm trying to get the knack of the language that they use in geometry class. Which, I will admit, that language kind of tends to disappear as you leave your geometry class. But since we're in geometry class, we'll use that language. So angle 2 is congruent to angle 3. Which means that their measure is the same. Or that they kind of did the same angle, essentially. Fair enough. So let's see. Statement one, angle 2 is congruent to angle 3. That's given, I drew that already up here. Statement two, angle 1 is congruent to angle 2, angle 3 is congruent to angle 4. So they're saying that angle 2 is congruent to angle 1. Or angle 1 is congruent to angle 2. That's right there. And that angle 4 is congruent to angle 3. Fair enough. And they say, what's the reason that you could give. And I forgot the actual terminology. But in my head, I was thinking opposite angles are equal or the measures are equal, or they are congruent. And you could just imagine two sticks and changing the angles of the intersection. You'll see that opposite angles are always going to be congruent. Let's see, that is the reason I would give. Opposite angles are congruent. Let's see which statement of the choices is most like what I just said. Complements of congruent angles are congruent. Complements. Vertical angles are congruent. I think that's what they call opposite angles. Vertical angles. I think that's what they mean by opposite angles. Supplements of congruent angles are congruent. That's not true. Corresponding angles are congruent. These aren't corresponding. I think this is what they mean by vertical angles. Complements of congruent angles are congruent. You know what, I'm going to look this up with you on Wikipedia. Let me see. Vertical angles. As you can see, at the age of 32 some of the terminology starts to escape you. What matters is that you understand the intuition and then you can do these Wikipedia searches to just make sure that you remember the right terminology. Let's see what Wikipedia has to say about it. Vertical angles. A pair of angles is said to be vertical or opposite, I guess I used the British English, opposite angles if the angles share the same vertex and are bounded by the same pair of lines but are opposite to each other. Right. So somehow, growing up in Louisiana, I somehow picked up the British English version of it. Maybe because the word opposite made a lot more sense to me than the word vertical. With that said, they're the same thing. Wikipedia has shown us the light. And so my logic of opposite angles is the same as their logic of vertical angles are congruent. Next problem. I'll start using the U.S. terminology. Although I think there are a good number of people outside of the U.S. who watch these. So maybe it's good that I somehow picked up the British English version of it. Once again, it might be hard for you to read. I'll read it out for you. Two lines in a plane always intersect in exactly one point. Fair enough. Which of the following best describes a counter example to the assertion above. I like to think of the answer even before seeing the choices. So can I think of two lines in a plane that always intersect at exactly one point. Well, what if they are parallel? What if I have that line and that line. They're never going to intersect with each other. They're parallel. That's the definition of parallel lines. The other example I can think of is if they're the same line. I guess you might not want to call them two the lines then. But I would. This line and then I had this line. Well that's parallel, but imagine they were right on top of each other, they would intersect everywhere. So either of those would be counter examples to the idea that two lines in a plane always intersect at exactly one point. And if we look at their choices, well OK, they have the first thing I just wrote there. Parallel lines, obviously they are two lines in a plane. But they don't intersect in one point. Problem eight. I'm going to make it a little bigger from now on so you can read it. OK, this is problem nine. Which figure can serve as the counter example to the conjecture below? If one pair of opposite sides of a quadrilateral is parallel, then the quadrilateral is a parallelogram. So once again, a lot of terminology. And I do remember these from my geometry days. Quadrilateral means four sides. A four sided figure. And a parallelogram means that all the opposite sides are parallel. For example, this is a parallelogram. Let me see how well I can do this. If you ignore this little part is hanging off there, that's a parallelogram. And if all the sides were the same, it's a rhombus and all of that. But that's a parallelogram. And that's a parallelogram because this side is parallel to that side. And this side is parallel to that side. All the sides are parallel. Now they say, if one pair of opposite sides of a quadrilateral is parallel, then the quadrilateral is a parallelogram. So I want to give a counter example. Let me draw a figure that has two sides that are parallel. Let's say that side and that side are parallel. And I don't want the other two to be parallel. Then it wouldn't be a parallelogram. Let's say the other sides are not parallel. Let's say they look like that. And like that. And once again, just digging in my head of definitions of shapes, that looks like a trapezoid to me. So let's see. Yeah, good, you have a trapezoid as a choice. Trapezoid. All the rest are parallelograms. A rectangle, all the sides are parellel. And we have all 90 degree angles. Rectangles are actually a subset of parallelograms. Rhombus, we have a parallelogram where all of the sides are the same length. All the angles aren't necessarily equal. Square is all the sides are parallel, equal, and all the angles are 90 degrees. So all of these are subsets of parallelograms. This is not a parallelogram. Although it does have two sides that are parallel. So this is the counter example to the conjecture. I think you're already seeing a pattern. In a lot of geometry, the terminology is often the hard part. The ideas aren't as deep as the terminology might suggest. Given, TRAP, that already makes me worried. Given TRAP is an isosceles trapezoid with diagonals RP and TA, which of the following must be true? OK, let's see what we can do here. So an isosceles trapezoid means that the two sides that lead up from the base to the top side are equal. Kind of like an isosceles triangle. So let me draw that. Actually, I'm kind of guessing that. I haven't seen the definition of an isosceles triangle anytime in the recent past. An isosceles trapezoid. But it sounds right. So I'll go with it. And that's a good skill in life. So I think what they say when they say an isosceles trapezoid, they are essentially saying that this side, it's a trapezoid, so that's going to be equal to that. They're saying that this side is equal to that side. An isosceles trapezoid. And they say RP and TA are diagonals of it. Let me draw that. So let me actually write the whole TRAP. So this is T R A P is a trapezoid. Let me draw the diagonals. RP is that diagonal. And TA is this diagonal right here. OK. All right, let's see what we can do. Which of the following must be true? RP is perpendicular to TA. Well, I can already tell you that that's not going to be true. And you don't even have to prove it. Because you can even visualize it. If you squeezed the top part down. Imagine some device where this is kind of a cross-section. If you were to squeeze the top down, they didn't tell us how high it is. Then these angles, let me see if I can draw it. That angle and that angle, which are opposite or vertical angles, which we know is the U.S. word for it. Those are going to get smaller and smaller if we squeeze it down. And in order for both of these to be perpendicular those would have to be 90 degree angles. And we already can see that that's definitely not the case. All right. RP is parallel to TA. Well that's clearly not the case, they intersect. All right, they're the diagonals. RP is congruent to TA. Well, that looks pretty good to me. Because it's an isosceles trapezoid. If we drew a line of symmetry here, everything you see on this side is going to be kind of congruent to its mirror image on that side. So both of these lines, this is going to be equal to this. And I can make the argument, but basically we know that RP, since this is an isosceles trapezoid, you could imagine kind of continuing a triangle and making an isosceles triangle here. Then we would know that that angle is equal to that angle. Well, actually I'm not going to go down that path. But you can almost look at it from inspection. Although, maybe I should do a little more rigorous definition of it. But RP is definitely going to be congruent to TA. Because both sides of these trapezoids are going to be symmetric. And so there's no way you could have RP being a different length than TA. Since this trapezoid is perfectly symmetric, since it's isoceles. And then D, RP bisects TA. Let's say if I were to draw this trapezoid slightly differently. If it looks something like this. If this was the trapezoid. This is also an isosceles trapezoid. And then the diagonals would look like this. So here, it's pretty clear that they're not bisecting each other. In order for them to bisect each other, this length would have to be equal to that length. And that's clear just by looking at it that that's not the case. That is not equal to that. So they're definitely not bisecting each other. So you can really, in this problem, knock out choices A, B and D. And say oh well choice C looks pretty good. But you can actually deduce that by using an argument of all of the angles. Anyway, that's going to waste your time. But that's a good exercise for you. Is to make the formal proof argument of why this is true. Although, you can make a pretty good intuitive argument just based on the symmetry of the triangle itself. Anyway, see you in the next video.