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CA Geometry: Similar triangles 1

10-14, similar triangles. Created by Sal Khan.

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Video transcript

Actually, just right after stopping that video, I realized a very simple way of showing you that RP is congruent to TA, a little bit more of a rigorous definition. If we can show that this triangle right there, that one I drew in purple, and this triangle right here are congruent, then we could make a fairly reasonable argument that RP is going to be congruent to TA, because they're essentially the corresponding sides of the two congruent triangles. This congruent triangles would be kind of flipping each other. So how can we make that argument? Well, on the purple triangle, this angle is going to be equal to this angle on the yellow triangle. Actually, we got that from the fact that this is an isosceles trapezoid, so the base angles are going to be the same. They told us this is an isosceles trapezoid, so we know that this side, right there, is going to be congruent to this side. And then finally, they both share this side right here. So we could use the argument-- once again, side, angle, side-- that the side, angle, and side are congruent to this side, angle, and side. So you could say by SAS, triangle TRP is congruent to triangle TAP. And if they're congruent, then all of the corresponding sides are equal, so then TA is congruent to RP. Once again, you didn't have to do all that. It's a multiple choice test. But I wanted to give you that. I felt bad I wasn't giving you a more rigorous definition. A more rigorous proof. So anyway, problem number 11. A conditional statement is shown below. If a quadrilateral has perpendicular diagonals, then it is a rhombus. Fair enough. Which of the following is a counterexample to the statement above? So they're saying, if it's perpendicular diagonals, then it's a rhombus. So if we could find something that has perpendicular diagonals that is not a rhombus, then we have a counterexample. Then this would not be true. So let's find something with perpendicular diagonals that is not a rhombus. Well, this one has perpendicular diagonals. The diagonals are perpendicular to each other, all 90-degree angles. And this is clearly not a rhombus. This is like a kite. This is not parallel to this and that is not parallel, so this is not a rhombus. So this is definitely a counterexample. This one does have perpendicular diagonals, but it's also a rhombus. So it's not a counterexample. It's just an example of what they're trying to say. This has perpendicular diagonals, it's a square, but a square is a subset of rhombuses. So this is another example. And, of course, this one does not have perpendicular diagonals. This is not a right angle. So A is the counterexample. Next question. Problem 12. Which triangles must be similar? Two obtuse triangles. Well, obtuse just means that they have two angles, one obtuse angle might look like that where that is greater than 90 degrees there, and the other obtuse might be super obtuse. It might be like that. And clearly these aren't similar. Well, this angle is obviously larger than that one. OK, so this is not similar. Similar means all the angles are the same. So it's like congruent, but you can scale them in size. That's how I think of them. Like this triangle. I'm trying to draw it so it looks exactly the same. That triangle. Well no, but you can imagine if I cut and pasted that, right? It would only be similar to this triangle if I drew everything to scale. Because the sides are different sizes, but all the angles are the same. That's what similar means. So let's see. Two scalene triangles with congruent bases. Well no, that's not true. That's not similar. Let's say they share the same base. One scalene triangle might look like this. It might come out a little bit and then go down like that. And the other scalene triangle has the same base right there. The sides might be a little closer to each other. So clearly, these two things aren't similar. This angle is different than that angle. All the angles are different, so they're not similar triangles. So that's not right. Two right triangles. Do those have to be similar? Well, no. You could have a right triangle that looks like this, where maybe the two sides are equal, right? That's a 45-45-90 triangle. Or you could have something like this where you have a 30-60-90 triangle. These clearly are not similar. All the angles are not the same. They both have a 90-degree angle. So I'm already guessing that D is our right answer, but let's see how it works out. Two isosceles triangles with congruent vertex angles. So I'm assuming when they say congruent vertex angles, I'm assuming they mean all of the angles are congruent. So two isosceles triangles. Let me think about it a little bit. Oh, actually, I think what they mean is angle in the middle when they say vertex angle. If that's one of my isosceles triangles. And isosceles triangle means that that side is equal to that side and that angle is equal to that angle. The vertex angle, I'm guessing, they mean is this angle right there. So if I had another isosceles triangle. Let's say maybe it's a little bit smaller. It looks something like that. And their vertex angles are the same. That angle is equal to this angle. Well, if that angle is equal to this angle and we know it's isosceles, so if we know it's isosceles, that is equal to that, then that has to be equal to that, we know that all the angles are the same. How do we know that this angle is equal to this angle? Well, think about it. Whatever angle this us, let's call this x. Let's call this angle y and this angle y. We know that x plus 2y is equal to 180, or that 2y is equal to 180 minus x, Or y is equal to 90 minus x over 2. Now if this is x, And let's call these z and z, So we know that x plus 2z is equal to 180. All the angles in a triangle have to add up to 180. Subtract x from both sides, you get 2z is equal to 180 minus x. Divide by 2, you get z is equal to 90 minus x over 2. So z and y are going to be the same angles. So all the angles are the same, so we're dealing with similar triangles. So choice D was definitely correct. 13. OK. Which of the following facts would be sufficient to prove that triangles ABC, that's the big triangle, and triangle DBE, so that's a small one, are similar? So we have to prove that all of their angles are similar. I cannot even look at the choices and I can guess where this is going. So we want to prove that those are similar. So first of all, they share the same angle. Angle ABC, this angle, is the same as angle DBE. So they share that same angle. So we got one angle down. Now let's think about it. If we knew that this angle is equal to that angle and that angle is equal to that angle, we'd be done. And the best way to come to that conclusion is if they told us that this and this are parallel. I'm guessing that's where they're going. Now I might have gone on a completely wrong tangent. Because it those two are parallel, then these two lines are transversals of the parallel. So that angle and that angle would be corresponding angles, so they would be congruent, and then that angle and that angle would be corresponding angles so they'd also be congruent. So if they told us that these are parallel, we're done. These are definitely similar triangles. And sure enough, choice C, they tell us that AC and DE are parallel. These are parallel, that's a transversal, this is a corresponding angle of that, so they're congruent. This is a corresponding angle to this, congruent, so all of the angles are congruent. So we have a similar triangle. Problem 14. OK. Parallelogram ABCD is shown below. Fair enough. Parallelogram: that tells us that the opposites sides are parallel. That's parallel to that, and then this is parallel to that. And all of the choices got clipped at the bottom, but I'll copy them over. Maybe I'll copy them above the question. Well, let me see what I can do. I think that's good enough. A little unconventional. OK. Parallelogram is shown below. They say which pair of triangles can be established to be congruent to prove that angle DAB is congruent to angle BCD? So DAB is this. Let me do it in another color. DAB is that angle, is congruent to BCD. They want us to show that those have the same angle measure. OK, and what do we have to show? They say what pair of triangles can be established to be congruent to prove that. OK, if these are both part of two different congruent triangles and they are the corresponding angles, then we know that they're congruent and we'd be done. So let's see what they say. Triangle ADC and BCD. BCD has this angle in it. BCD does help us because it has this angle in it, but triangle ADC does not have this angle in it, right? Triangle ADC has this the smaller angle in it. ADC doesn't involve this whole thing, so that's not going to help us. Triangle AED, once again, does not involve this larger angle, does not involve the angle DAB. It only involves the little smaller angle, so that's not going to help us. Triangle DAB. That looks good. That has this whole angle in it. DAB. And then BCD. If we showed that that triangle is congruent to this triangle right here, I think we're done. That would be enough to show that this angle is congruent to that angle, because they would be the corresponding angles of a congruent triangle. So I think C is where we're going to go. Let's just look at choice D. DEC. Once again, triangle DEC. Let me make this point clear. Triangle DEC does not involve either of the angles we care. It clearly does not involve this angle, and it only involves part of this angle, only this part. It doesn't involve this whole angle, so that's not going to help us either. So the answer is C. Anyway, see you in the next video.