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# CA Geometry: More on congruent and similar triangles

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All right. We're on problem 17. It says, which of the following
best describes the triangles shown below? OK, they want to know,
are the similar? Are they congruent? Et cetera. They tell us that this
is a 60 degree angle. This is a 90 degree angle, they
do this little square thing there. The angles in a triangle
have to add up to 180. So if this is 90, this is
60, that adds up to 150. So this has to be
180 minus 150. So this has to be a
30 degree angle. So that has to be 30 degrees
right there. Fair enough. Now let's get this one. That's 30, that's 90. Well, by the same argument,
this has got to be 60. Because they all have
to add up to 180. All right. So just there we know that all
of the angles in both of the triangles are congruent. Or that their measures
are equal. So we know already that
these are definitely both similar triangles. Now, a similar triangle also
tells us that the ratio of all of the sides are equal. So. if you just sort of eyeball it,
if you said, OK, the side opposite the 90 degree, these
are the corresponding sides, the ratios are equal. But we see that they give
us the actual lengths. So the hypotenuse of both
of these triangles is 8. So the ratio is actually 1:1. And when the ratio of the sides
is 1:1, when the sides are actually congruent, and if
you're just given one side, that's enough. Then you could actually figure
out the rest of them using a little trigonometry or
something like that. We're not going to go
there just yet. But in geometry class, you
learned that if something is similar and and at least one of
the corresponding sides is congruent, then the whole thing is going to be congruent. So these are both similar
and congruent triangles. Both similar and congruent,
that's A. Problem 18. OK, let me cut and paste it. Which of the following
statements must be true if triangle GHI is similar. So if they write this that
means congruent. If they just write that,
that means similar. Which of the following
statements must be true if triangle GHI is similar
to triangle JKL? So even before looking at the
choices, that means that the ratio of all the sides are
the same, or all of the angles are the same. Let's see what they give us. The two triangles
must be scalene. Now, you have similar triangles
that are isosceles or equilateral. That's not right. The two triangles must have
exactly one acute angle. No, they could have
two acute angles. They could have three
acute angles. The way that they've drawn
it here, actually all of them are acute. None of these angles are greater
than 90 degrees just the way they've drawn. So that's not right. Some of these statements
are so crazy that they're hard to process. Anyway, C, at least one of the
sides of the two triangles must be parallel. I don't care how they're
oriented. You don't care about the
orientation of the triangles. The corresponding sides of
the triangles must be proportional. Yeah, that's one of the ways
that you know that something is similar. That the corresponding sides
are proportional. So that is, D. So this is almost, do you
know the definition of a similar triangle? Question 19. Let me erase this. OK. I have copied it. Now I am pasting it. In the figure below, AC
is congruent to DF. OK, so they're equal
to each other. AC and DF are congruent. And angle A is congruent
to angle D. Fair enough. That's angle A, that's
angle D. That's what they tell us. Which additional information
would be enough to prove that triangle ABC is congruent
to DEF? So they just gave us one
side and one angle. If they gave us another side,
if they said that DE is congruent to AB, that'd
be pretty cool. If they gave us this angle, if
they said angle F is congruent to angle C, that'd be good. Let's see what they give us. AB is congruent to DE. Yeah, sure. If AB is congruent to DE
then we definitely have congruent triangles. And you know the theorem that
you would have to say in your geometry class is, I have a
side, an angle, and a side. So you would say by SAS, by
side, angle, side, I know that these two triangles
are congruent. So AB is congruent to DE. Let's look at the other
ones to make sure we didn't miss anything. AB is congruent to BC. Well, that's fine. But that doesn't tell us
how AB relates to DE. So that's a useless statement. BC is congruent to EF. Well, see, this is another
time that I have a slight problem with the way they're
going with this. Because if BC were
congruent to EF. Let me think about that. Could I draw this triangle
in a way where they're still not congruent. Because I have this angle
here constraining it. They told us that. So it's not like I can
draw this line, FE, coming out here. Because if it came out here,
then DE would have to come like that. And then this angle couldn't
be what they said it was. So I'm just trying to think, I
actually think that would be sufficient. If you're given that this side
is congruent to that side. I think you could make a
trigonometric argument very easily to show that these two
triangles have equal sides. But anyway, I'm not going
to bother with that. Let's see. Let's look at choice D. BC is congruent to DE. Well, these aren't even
corresponding sides. So that's clearly useless. I have a suspicion that
this would have also been enough to prove. But anyway, I don't want
to insult anyone in the California Department of
Education, but I'm slightly disappointed by some
of these questions. Because I feel like they
really aren't testing intuition, they're just testing
to see whether you know the definitions of some of
these geometric terms. And whether you can spout out, side,
angle, side, and angle, side, angle. And things like that. And you're going to forget those
about three hours after you take the test. That's
pretty useless. What's useful is if you know
something that gives you an intuition about triangles. That's going to be useful for
you on the SAT, that's going to be useful for you when
you take trigonometry. And I'll tell you
a dirty secret. You will never use ASA theorem
or SAS theorem or anything like that again in your
mathematical careers. Your 9th or 10th grade geometry
class is the first and the last time that
you'll ever see them. So I have a slight problem
where they want you to memorize these theorems
and all of that. And even some of this notation
never shows up again in your mathematical careers. Even if you do a PhD
in mathematics. The only time you'll probably
see it again is if you become a geometry math teacher. Anyway, but it's good. I mean you should know how to do
this stuff at minimum just to jump through that
loop that society makes us all jump through. So problem 20. You don't want someone else to
get paid more just because they were willing
to say SAS, ASA. Anyway, all right, problem 20. Given AB and CD intersect
at point E. And just another aside, I think
you can even tell from my tone that I enjoy the SAT
problems a lot more. Because in some ways, in fact,
in every way, the SAT problems really test your understanding
of geometry, but never do they mention the words similar,
congruent, SAS, ASA. They never mention all these
things that you essentially memorize in your
geometry class. And I know tons of people who
get A's in geometry and then they don't do well on the SAT. And I know people who
do the other way. And frankly, I'd rather
hire the person who does well on the SAT. Because that's the person who
I think has the intuition. But anyway, we have
to do this. And I probably shouldn't
rant like that. Given AB and CD intersect
at point E. Fair enough. And they tell us that angle
1 is congruent to angle 2. So that and that are equal. All right, so already
those look like alternate interior angles. If this line were parallel. In fact, I think that's enough
to show that this line is parallel to this line. Because, if you view this as a
transversal, if you view DC as a transversal, then you see
that's a transversal between these two lines. And because the alternate
interior angles are the same, or they're congruent, you know
that those are going to be parallel lines. But anyway, I don't know if
that's at all useful. What are they going to ask us? Which theorem or postulate can
be used to prove that AED is similar to BEC. OK. So let's see, so I didn't
have to even say those are parallel lines. So what do they tell us? First of all, we know that 3
and 4 are congruent angles, because they're opposite
angles. Once again, I don't like the
word vertical angles, because these angles are clearly
not vertical. They're more side by side. But they're definitely
opposite. So those two angles
are the same. 1 and 2 are the same, 3
and 4 are the same. If you know two of the
angles in a triangle, you know the third. So this angle and that angle
have to be the same. But in general, if you know that
two angles of a triangle are the same, the third
has to be the same. So, that tells you that it's
a similar triangle. So we could use angle, angle. We know that two angles are the
same as two other angles. So we know we're dealing
with similar triangles. Anyway, all out of time
because of my rant. See you in the next video.