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# Missing angles (CA geometry)

Video transcript

We're on problem 46. In the figure below, line
segment AB is parallel to C. I think there's a typo here. I've got to believe this
has got to be CD. That's got to be a typo. All right, so AB, I'm assuming,
is parallel to CD. That's probably what they wanted
to have, because you can't be parallel to a vertex. And they say, what is
the value of x? So the way to think about it is,
if these two are parallel lines, then this segment up
here is a transversal. If we just extend the two lines
like this, let me see if I can do it well. So if I extended that line like
that, and if I extended this line like that, then
segment AD is just a transversal. Let me see if I can
do that one right. So if I just extended it like
that, you see it's a transversal, and I could go the
other direction as well. I think you get the idea. So if that is a transversal,
what do you know about transversals? We know that this angle right
here, I'll draw it small, is congruent to this
angle over here. So the measure of this angle
is also x plus 40. Because they're corresponding
angles, and you could see that by inspection, and if you moved
around the transversal, it would make sense that
that's the case. So this is x plus 40 and this
is minus 40, and they're clearly supplements
of each other. They're supplementary angles. Then the sum of these
two angles have to be equal to 180. So let's figure it out. So x minus 40 plus x plus 40
is equal to 180, because they're supplementary. The 40's cancel out. So you just minus 40, plus
40 adds up to zero. So you're left with 2x
is equal to 180. x is equal to 90. So it's D. 47: The measures of the interior
angles of a pentagon are 2x, 6x, 4x minus 6, 2x
minus 16, and 6x plus 2. What is the measure in degrees
of the largest angle? OK, so first of all, we have to
remember what is the sum of the interior angles
of a pentagon? And that's where I always draw
an arbitrary pentagon. Let me see if I can do that. Actually, there's a
polygon tool here. How does it work? I'm just trying to
draw a pentagon. I don't know if that's any
different than the line tool, but regardless. So how many triangles can
I draw in a pentagon? And that tells me what my total
interior angles are. And there is a formula for that,
but I like relying on your reasoning more than the
formula, because you might forget the formula, or even
worse, you might remember it, but not have the confidence to
use it, or you might remember it wrong ten years
in the future. So the best thing to do, if
you have a polygon, is to count the triangles in it. Straightforward enough. It's almost easier than
using the formula. So a pentagon has three
triangles in it. So the sum of its interior
angles are going to be 3 times 180, because it has
3 triangles in it. Each triangle has 180 degrees. And I know you can't see
what I just wrote. So the sum of all of these
angles are going to be the sum of all of the angles in
all three triangles. So it's 3 times 180 is
equal to 540 degrees. So that's the sum of all of
these interior angles. Now, they say that each of them
are 2x, 6x, et cetera. So the sum of all of these terms
have to be equal 540. I'm going to write
them vertically. It makes them easier to add. So if we write there 2x, 6x,
4x minus 6, 2x minus 16, and 6x plus 2. This is going to be the
largest, right? That sum is going
to equal 540. So let's add this up. Minus 6, minus 16,
that's minus 22. Plus 2 is minus 20. That's right. And 2x plus 6x that is 8x plus
4 is 12, 12 plus 2 is 14, 14 plus 6 is 20. So we have 20x minus 20 is
equal to 540 degrees. Let me write it again. 20x minus 20 is equal to 540. Let's divide both sides of
this equation by 20. So you get x 1 minus 1 is equal
to-- it would be 54 divided by 2, which
is equal to 27. Add 1 to both sides,
x is equal to 28. And they want to know, what is
a measure in degrees of the largest angle? That's going to be this one. That's the largest one. It's 6 times x plus 2. So 6 times 28, that's 48. 2 times 6 is 12 plus 4 is 168. So it's 168 plus 2. It's 170 degrees. Choice C. Problem 48: What is the
measure of angle 1? So this we're going into
the angle game. And these are fun, because
they are kind of these deductive reasoning problems
where you just use a couple of simple rules and just fill
in the whole thing. So let's think about it. This is 36 degrees. They tell us that this
whole angle right here is a right angle. So this angle right here is
going to be the complement to 36 degrees. 36 plus this angle have
to be equal to 90. So what's this one? This one is 90 minus
36, which is 54. That's going to be 54 degrees. 90 minus 30 is 60. Right, that's 54. And this angle right here,
that's going to be the supplement of 88. So this is going to be-- I'll
do it in a different color-- 180 minus 88. That is equal to 92 degrees. Now this angle 1 plus the 54
plus the 92 is equal to 180. So we know that-- let's say
angle one plus 54 plus 92 is equal to 180. This is 146 is equal to 180. Subtract 146 from both sides. The measure of angle one is
equal to 80 minus 40 is 40, so 80 minus 46 is equal
to 34 degrees. So the answer is A. Problem 49: What is the
measure of angle WZX? So they want to know what this
angle right here is. Let's do the angle
game some more. Let's see, we can immediately
figure out what this angle is, because it is the supplement
of 132 degrees, so this is going to be 180 minus 132. So this is 48 degrees. This angle plus this angle
is going to be equal to this angle. Or this angle plus this
angle plus this angle is equal to 180. I don't know what I just
said, I think I said something wrong. Write that down. So this angle is going
to be equal to 180 minus 52 minus 48. Because the sum of the angles
add up to 180, and so that is equal to 180 minus 100 which
equals 80 degrees. So this angle right here
is equal to 80 degrees. And the angle they want us to
figure out is the opposite of this angle, or in the
U.S., I guess, they say vertical angles. And so opposite or vertical
angles are equal or they're congruent, so this is going
to be 80 degrees as well. And that is choice A. Problem 50: What is the measure
of an exterior angle of a regular hexagon? A regular hexagon tells us that
all of the sides are the same, it's equilateral, and
all of the angles are the same, equiangular. So if we just knew what's the
total degree measure of the interior angles, we could just
divide that by 6, and then that would give us what each
of the interior angles are, and then we could use that
information to figure out the exterior angles. Let's just do it. So once again, I like to
just draw a hexagon. Let's just draw a hexagon and
count the triangles in it. Two sides, three sides,
four sides, five sides and six sides. And how many triangles
do I have here? One, two, three. So I have one, two, three,
four triangles. The sum of the interior angles
of this hexagon, of any hexagon, whether it's regular
or not, are going to be 4 times 180 and that's
720 degrees. And it's a regular hexagon, so
all the interior angles are going to be the same. And there's six of them. So each of them are going
to be 720 divided by 6. Well, 6 goes into
72 twelve times. So each of the interior angles
are going to be 120 degrees. And I didn't draw it that
regular, but we can assume that all of these are
each 120 degrees. Fair enough. Now, if all of those are each
120 degrees, what is the measure of an exterior angle? Well, we could just extend
one of these sides out a little bit. We could say, OK, if this is
120 degrees, what is its supplement? Well, these have to add
up to 180, so 180 minus 120 is 60 degrees. I could do it on any side. I could extend that line out
there, and I'd say, oh, that's 60 degrees. So any of the exterior angles
are 60 degrees. B. All right, do I have
time for one more? I'll wait for the next one
in the next video. See you soon.