Learn about CA Geometry: More Proofs

All right, we’re on problem number seven and when I copy and paste it I made 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 the angle 2 is congruent to angle 3. So, angle 2 or 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 used in Geometry class, which I will admit, that language kind of intends to disappear as you leave your Geometry class.

But, since we’re in geometry class, we’ll use that language. So, the angle 2 is congruent to angle 3 which means that their measure is the same or that they kind of they’re the same angle, essentially. But fair enough. So, let’s see.

Statement 1: angle 2 is congruent to angle 3. That’s given. I drew that already up here. Statement 2: 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 there right there. And then, angle 4 is congruent to angle 3. Fair enough. And they say, “What’s the reason that you could give?” And I don’t know what—

I forgot the actual terminology but I had in my head I was thinking, “Oh, opposite angles are equal or the measures are equal or they’re congruent, right? And you could just imagine two sticks and chase the angles of the intersection. You’ll see that opposite angle are always going to be congruent. But let’s see. That is the reason that 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. Supplement 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. When the supplement of congruent angles is congruent, that’s not true. Corresponding angles are congruent. This on 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 terminologies start to escape you. What matters is that you understand the intuition and that you can do this Wikipedia search 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 that’s said to be vertical or opposite. Oh, 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— next problem after using the US terminology— Oh, I think there are a good number of people outside of the US who watch this. So maybe, it’s good that I somehow picked up the British English version of it. Okay.

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? So I’d like to think of the answer even forcing the choices. So can I think of two lines in a plane that always intersect in exactly one point? Well, what if they’re parallel, right? What if you know, I have— you see 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 mean I guess you might not want to call them two lines then but you know I would. This line and then I had you know, well that’s parallel but imagine if they’re 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 in exactly one point. And if we look at their choices, well, okay, they have the first thing I just write there parallel lines. Obviously, there are two lines in a plane, but they don’t intersect in one point.

Problem eight: I’m going to make it write a little bigger from now on just so you could read it. Okay, this is problem nine. Problem nine: Which figure can serve as a counter example to the conjecture below? If one pair of opposite sides of a quadrilateral is parallel, 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, right? A four sided figure. And a parallelogram means that all the opposite sides are parallel.

So for example, this is a parallelogram. I don’t know if you remember what— you know. This let me see if I can— how well I can do this. Then like— well that’s if you ignore this little part that’s 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 parallel because— 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 the opposite sides of a quadrilateral is parallel, then the quadrilateral is a parallelogram. So, I’m going to give a counter example. So let me make one – draw figure that has two sides that are parallel. So let’s say that side and that side are parallel. And then, I don’t want the other two to be parallel. Then it wouldn’t be a parallelogram.

Now, let say the other sides are not parallel. So, they look like let’s say, they look like that and like that. And once again, that— you know, just digging in my head of definitions of shapes. That looks like a trapezoid to me. So let’s see. Yeah. Good. We have a trapezoid is a choice— trapezoid. All the rest are parallelograms, right? A rectangle is all the sides are parallel. And we have all 90 degree angles, right? Rectangles are actually a subset of parallelograms. Rhombus is— we have a parallelogram where all of the sides are the same length. All the angles are necessarily equal. Square is all the sides are parallel, equal and all the angles are 90 degrees. So, all of these are subsets of parallelogram. This is not a parallelogram, although it does have two sides that are parallel. So, this is the counter example to the conjecture.

I thing you’re already seeing the pattern, the lot of Geometry. The terminology is often the hard part. The ideas aren’t as deep as the terminology might suggest. Okay. Given TRAP— that already makes me worried. All right, given a trapezoid as an isosceles trapezoid with diagonal RP and TA, which of the following must be true? An isosceles tri— Okay, let’s see what we can do here.

So— an isosceles trapezoid means that the two sides on the two kinds of 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 any time of the recent past but, an isosceles trapezoid. But it sounds right. So I’ll go with it. And it’s a good skill in life. Whoops! To make your best, let’s see.

So, I think when they say isosceles trapezoid, they’re essentially saying that this side— you know it’s a trapezoid. So, that’s going to be equal to that. And they’re saying that this side is equal to that side— so, isosceles trapezoid. Okay? Which of the following— and they say RP and TA are diagonals of it. So, let me draw that. So, let me actually write the whole TRAP. So, this is TRAP is a trapezoid. And when we draw the diagonals RP is that diagonal and TA is this diagonal right here. Okay. 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 this R— if you squeezed the top part down, right? Imagine some device with this kind of the cross section. If you would squeeze the top down, right? They didn’t tell us how high. Then these angles, you can imagine that both of these angles—.

Let me just see if I could draw it. That angle and that angle which are opposite or vertical angles, which you know, that’s the US word for. Those are going to get smaller and smaller if we squeeze it down, right? An in order for 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. They’re the diagonals. RP is congruent to TA. Well now, that looks pretty good to me, right? Because it’s an isosceles trapezoid, you can— this whole— 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, right? So both of these lines, this is going to be equal to this and I could make the argument, but basically, we know that RP— since this is an isosceles trapezoid, you could imagine the kind of continuing a triangle and making an isosceles triangle here. Then we would know that that angle is equal to that angle. Let see.

We could make a— you see that angle is equal to that angle and well, actually, I’m not going down that path because I think that will be— but you can— I mean, you can almost look at it from inspection, although maybe I should be doing a little more rigorous definition of it. But RP is definitely going to be parallel. It’s definitely going to be congruent to TA because both sides of this trapezoid are going to be symmetric and so, there’s no way you could have RP being a different length than TA, right? Since this trapezoid is perfectly symmetric, since it’s isosceles— and then the RP bisects TA. RP bisects TA. And again, you can show that, let’s say if I were to draw this trapezoid slightly differently - if it looks something like this. Let me draw it like this.

If this was a trapezoid, then this is also an isosceles trapezoid. And then the diagonals would look like this. Whoops! The diagonals would look like this. So here, it’s pretty clear that they’re not bisecting each other, right? It’s pretty— in order 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, right? 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 just saying— by using an argument of all of the angles that this angle is going to be equal to— let me see if I can make the argument.

Anyway, that’s going to waste your time. But that’s a good exercise for you is to make a formal proof of 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.