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CCSS.Math:

So we have two
intersecting lines here, and then we have this
other purple-looking line. And then they've
given us some angles. They tell us that the measure
of this angle right over here is 7x. The measure of this angle
right over here is 60 degrees. And the measure of this
angle right over here is x. So let's try to figure out
what all of these angles are. So to do that, we have
to figure out what x is. And there's a big clue here,
because the 60-degree angle plus the x angle,
they're adjacent. And if you add these two angles
together, their outer rays, our vertical angle
with this 7x angle. So we could say-- and just to
visualize that a little better, let me color it in. Actually, let me do it this way. You see that this
angle out here-- let me do it in a color
I haven't used yet. This entire angle
over here, which is going to be 60
degrees plus x, that's a vertical angle with
this angle, the one that has measure 7x. So we could say that 60
degrees plus x is equal to 7x because vertical
angles are equal. So let's write that down. We get 60. And we'll assume that
everything is in degrees. 60 plus-- let me do that
in this other color. 60 plus x is going
to be equal to 7x. And now we just
have to solve for x. So the simplest
thing to do would be to get all over x's on
one side of the equation. I've already gotten seven
x's on this right-hand side, so let's get rid of all of
the x's on the left-hand side. And the easiest way
to get rid of this x is to subtract x from
the left-hand side. But of course, in order
to keep it an equation, we can't just do
something to one side. Otherwise, it won't
be equal anymore. We have to do it to both sides. So let's subtract
x from both sides. And on the left-hand side,
we are left with just the 60. So we're left with just a 60. And then that is going to
be equal to 7x minus x. If I have 7 of something
and I get rid of 1 of them, I'm going to have 6 of
that something left. So that's going
to be equal to 6x. So we have 6 times
something is equal to 60. You could probably figure
that out in your head. But I will do it a
little systematically. We can divide both sides
by 6 to solve for x. So let's do that. And we would be left
with x is equal to. 60 divided by 6 is 10. And we reminded ourselves that
everything was in degrees. And we could even do that here. This was in degrees. This is in degrees. And so this is in
degrees right over here. So the measure of this angle
right over here is 10 degrees. So this one right over
there is 10 degrees. This is, of course, 60 degrees. You add them together, 60
degrees plus 10 degrees is 70 degrees. So this bigger angle right
over here is 70 degrees. And of course, this one over
here, it's a vertical angle. It's going to have
to be the same. And we see that, 7 times x. 7 times 10 degrees is
70 degrees as well.