Example using algebra to find measure of complementary angles

So we're told that ray OL
is perpendicular to ray ON.

So OL is perpendicular to ON.

So we know that this right
over here is a right angle,

or it's a 90-degree angle.

Then they tell us that
the measure of angle LOM

is equal to 2x plus 46.

So LOM-- so this angle
right over here--

is equal to 2x plus 46.

Then they tell us that
the measure of angle MON

is equal to 3x minus 6.

So this angle right
over here is 3x minus 6.

And so they ask
us find angle MON.

So we need to find this
angle right over here.

And we'd be able to find
it if we knew what x is.

If we knew what x is, then
it's going to be 3 times

that minus 6.

Now, the one thing
that we know is

that if we add up
these two angles,

these are adjacent angles.

And their outside rays
form a right angle.

So these two angles are going
to add up to 90 degrees.

And I guess we can assume here
that we're dealing in degrees.

So let's say that 2x
plus 46 plus 3x minus 6

is going to add
up to 90 degrees.

These two angles
are complementary.

So it's going to add
up to 90 degrees,

or it's going to add up to 90.

And now, we just
have to simplify.

We have 2 x's.

We have another 3 x's.

So you add those two, and
you're going to get 5 x's.

And then you have 46.

And you're going to subtract 6.

So it's going to be
plus 40 is equal to 90.

Subtract 40 from both sides.

So let's do that.

Subtract 40 from both sides.

And on the left-hand side,
you're just left with a 5x.

And on the right-hand side,
you're left with a 50.

Now, just divide
both sides by 5.

And we are left with
x is equal to 10.

Now, our answer is not 10.

We've just figured
out what x is.

What they ask us for is what's
the measure of angle MON?

Well, we already know that
the measure of angle MON

is equal to 3x minus 6, which
is equal to 3 times 10 minus 6--

I want to do that minus 6
in that green color-- which

is equal to 24 degrees.