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# Triangle angle challenge problem 2

Video transcript

Thought I would do some
more example problems involving triangles. And so this first one, it says
the measure of the largest angle in a triangle
is 4 times the measure of the second largest angle. The smallest angle
is 10 degrees. What are the measures
of all the angles? Well, we know one of them. We know it's 10 degrees. Let's draw an arbitrary
triangle right over here. So let's say that
is our triangle. We know that the smallest angle
is going to be 10 degrees. And I'll just say,
let's just assume that this right over here is the
measure of the smallest angle. It's 10 degrees. Now let's call the
second largest angle-- let's call that x. So the second largest
angle, let's call that x. So this is going to be x. And then the first
sentence, they say the measure of the
largest angle in a triangle is 4 times the measure of
the second largest angle. So the second
largest angle is x. 4 times that measure
is going to be 4x. So the largest angle
is going to be 4x. And so the one thing we
know about the measures of the angles
inside of a triangle is that they add
up to 180 degrees. So we know that 4x
plus x plus 10 degrees is going to be equal
to 180 degrees. It's going to be equal to 180. And 4x plus x, that
just gives us 5x. And then we have 5x plus
10 is equal to 180 degrees. Subtract 10 from both sides. You get 5x is equal to 170. And so x is equal to 170/5. And let's see, it'll go into
it-- what is that, 34 times? Let me verify this. So 5 goes into-- yeah,
it should be 34 times because it's going to go into
it twice as many times as 10 would go into it. 10 would go into 170 17 times. 5 would go into 170 34 times. So we could verify it. Go into 170. 5 goes into 17 three times. 3 times 5 is 15. Subtract, you get 2. Bring down the 0. 5 goes into 20 four
times, and then you're not going to have a remainder. 4 times 5 is 20. No remainder. So it's 34 times. So x is equal to 34. So the second largest angle
has a measure of 34 degrees. This angle up here is
going to be 4 times that. So 4 times 34--
let's see, that's going to be 120 degrees
plus 16 degrees. This is going to be 136 degrees. Is that right? 4 times 4 is 16, 4 times
3 is 120, 16 plus 120 is 136 degrees. So we're done. The three measures, or the
sizes of the three angles, are 10 degrees, 34
degrees, and 136 degrees. Let's do another one. So let's see. We have a little bit
of a drawing here. And what I want
to do is-- and we could think about
different things. We could say, let's solve for x. I'm assuming that 4x is
the measure of this angle. 2x is the measure of that
angle right over there. We can solve for x. And then if we know
x, we can figure out what the actual measures
of these angles are, assuming that we
can figure out x. And the other thing
that they tell us is that this line
over here is parallel to this line over here. And it was very craftily drawn. Because it's parallel,
but one stops here, and then one starts up there. So the first thing
I want to do-- if they're telling us that
these two lines are parallel, there's probably going
to be something involving transversals or something. It might be something
involving-- the other option is something
involving triangles. And at first, you might
say, wait, is this angle and that angle vertical angles? But you have to be very careful. They are not. This is not the same line. This line is parallel
to that line. This line, it's bending
right over there, so we can't make any type
of assumption like that. So the interesting
thing-- and I'm not sure if this will lead in
the right direction-- is to just make it
clear that these two are part of parallel lines. So I could continue this
line down like this. And then I can continue
this line up like that. And then that starts to look
a little bit more like we're used to when we're dealing
with parallel lines. And then this line
segment, BC-- or we could even say line BC, if
we were to continue it on. If we were to continue it
on and on, even pass D, then this is clearly
a transversal of those two parallel lines. This is clearly a transversal. And so if this angle
right over here is 4x, it has a corresponding angle. Half of the-- or maybe most
of the work on all of these is to try to see the parallel
lines and see the transversal and see the things that
might be useful for you. So that right there
is the transversal. These are the parallel lines. That's one parallel line. That is the other parallel line. You can almost try
to zone out all of the other stuff
in the diagram. And so if this angle
right over here is 4x, it has a corresponding
angle where the transversal intersects the
other parallel line. This right here is its
corresponding angle. So let me draw it
in that same yellow. This right over here is
a corresponding angle. So this will also be 4x. And we see that this angle--
this angle and this angle, this angle that has measure
4x and this angle that measures 2x-- we see that
they're supplementary. They're adjacent to each other. Their outer sides
form a straight angle. So they're supplementary,
which means that their measures
add up to 180 degrees. They kind of form-- they
go all the way around like that if you add the two
adjacent angles together. So we know that 4x plus 2x needs
to be equal to 180 degrees, or we get 6x is
equal to 180 degrees. Divide both sides by 6. You get x is equal to 30,
or x is equal to-- well, I shouldn't say--
well, x could be 30. And then this angle right
over here is 2 times x. So it's going to be 60 degrees. So this angle right over here
is going to be 60 degrees. And this angle right
over here is 4 times x. So it is 120 degrees,
and we're done.