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CCSS Math: 8.G.A.5, HSG.CO.C.10

What I want to do
now is just a series of problems that really make
sure that we know what we're doing with parallel lines and
triangles and all the rest. And what we have right here
is a fairly classic problem. And what I want to do is I
want to figure out, just given the information
here-- so obviously I have a triangle here. I have another
triangle over here. We were given some of the angles
inside of these triangles. Given the information
over here, I want to figure out what
the measure of this angle is right over there. I need to figure out what
that question mark is. And so you might want
to give a go at it just knowing what you know about
the sums of the measures of the angles inside of
a triangle, and maybe a little bit of what you know
about supplementary angles. So you might want to pause
it and give it a try yourself because I'm about to
give you the solution. So the first thing
you might say-- and this is a
general way to think about a lot of these problems
where they give you some angles and you have to figure out
some other angles based on the sum of angles and
a triangle equaling 180, or this one doesn't have
parallel lines on it. But you might see some
with parallel lines and supplementary lines
and complementary lines-- is to just fill in everything
that you can figure out, and one way or
another, you probably would be able to figure out
what this question mark is. So the first thing that
kind of pops out to me is we have one triangle
right over here. We have this
triangle on the left. And on this triangle
on the left, we're given 2 of the angles. And if you have 2 of the
angles in a triangle, you can always figure
out the third angle because they're going to
add up to 180 degrees. So if you call that x, we
know that x plus 50 plus 64 is going to be equal
to 180 degrees. Or we could say, x
plus, what is this, 114. X plus 114 is equal
to 180 degrees. We could subtract 114 from
both sides of this equation, and we get x is equal
to 180 minus 114. So 80 minus 14. 80 minus 10 would be 70,
minus another 4 is 66. So x is 66 degrees. Now, if x is 66
degrees, I think you might find that there's
another angle that's not too hard to figure out. So let me write it like this. Let me write x is
equal to 66 degrees. Well if we know this
angle right over here, if we know the measure of
this angle is 66 degrees, we know that that
angle is supplementary with this angle right over here. Their outer sides
form a straight angle, and they are adjacent. So if we call this
angle right over here, y, we know that
y plus x is going to be equal to 180 degrees. And we know x is
equal to 66 degrees. So this is 66. And so we can subtract
66 from both sides, and we get y is equal to--
these cancel out-- 180 minus 66 is 114. And that number might look
a little familiar to you. Notice, this 114 was
the exact same sum of these 2 angles over here. And that's actually
a general idea, and I'll do it on the side
here just to prove it to you. If I have, let's say that
these 2 angles-- let's say that the measure
of that angle is a, the measure
of that angle is b, the measure of
this angle we know is going to be 180
minus a minus b. That's this angle
right over here. And then this angle,
which is considered to be an exterior angle. So in this example, y
is an exterior angle. In this example, that
is our exterior angle. That is going to be
supplementary to 180 minus a minus b. So this angle plus
180 minus a minus b is going to be equal to 180. So if you call this
angle y, you would have y plus 180 minus a
minus b is equal to 180. You could subtract
180 from both sides. You could add a plus
b to both sides. So plus a plus b. Running out of space
on the right hand side. And then you're left
with-- these cancel out. On the left hand side,
you're left with y. On the right hand side
is equal to a plus b. So this is just a
general property. You can just reason
it through yourself just with the sum of the
measures of the angles inside of a triangle add
up to 180 degrees, and then you have a
supplementary angles right over here. Or you could just say, look,
if I have the exterior angles right over here,
it's equal to the sum of the remote interior angles. That's just a little
terminology you could see there. So y is equal to a plus b. 114 degrees, we've already
shown to ourselves, is equal to 64 plus 50 degrees. But anyway, regardless
of how we do it, if we just reason
it out step by step or if we just knew this
property from the get go, if we know that y is
equal to 114 degrees-- and I like to reason it out
every time just to make sure I'm not jumping to conclusions. So if y is 114 degrees,
now we know this angle. We were given this
angle in the beginning. Now we just have to figure
out this third angle in this triangle. So if we call this z, if we
call this question mark is equal to z, we know
that z plus 114 plus 31 is equal to 180 degrees. The sums of the measures of
the angle inside of a triangle add up to 180 degrees. That's the only property
we're using in this step. So we get z plus, what is
this, 145 is equal to 180. Did I do that right? We have a 15, then a 30. Yep, 145 is equal to 180. Subtract 145 from both
sides of this equation, and we are left with z is equal
to 80 minus 45 is equal to 35. So z is equal to 35
degrees, and we are done.