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CCSS Math: 7.G.B.5

Let's say I have an
angle ABC, and it looks something like this. So its vertex is going
to be at B. Maybe A sits right over here, and
C sits right over there. And then also let's say that we
have another angle called DBA. I want to have the vertex
once again at B. So let's say it looks like this. So this right over here is our
point D. That is our point D. And let's say that we know
that the measure of angle DBA is equal to 40 degrees. So this angle right
over here, its measure is equal to 40 degrees. And let's say that we know
that the measure of angle ABC is equal to 50 degrees. So there's a bunch of
interesting things happening here. The first interesting thing
that you might realize is that both of these
angles share a side. If you view these as rays-- they
could be lines, line segments, or rays-- but if you
view them as rays, they both share this ray BA. And when you have
two angles like this that share the same side, these
are called adjacent angles. Because the word "adjacent"
literally means next to. These are adjacent. They are adjacent angles. Now there's something
else that you might notice that's
interesting here. We know that the measure
of angle DBA is 40 degrees and the measure of
angle ABC is 50 degrees. And you might be able to guess
what the measure of angle DBC is. If we drew a
protractor over here-- I'm not going to draw it. It will make my
drawing all messy. Well, maybe I'll
draw it really fast. So if you had a protractor
right over here, clearly this is opening
up to 50 degrees. And this is going
another 40 degrees. So if you wanted to say what
the measure of angle DBC is, it would essentially be the sum
of 40 degrees and 50 degrees. And let me delete
all of this stuff right here to keep things clean. So the measure of angle DBC
would be equal to 90 degrees. And we already know that 90
degrees is a special angle. This is a right angle. There's also a word for
two angles whose sum add up to 90 degrees, and
that is complementary. So we can also say that
angles DBA and angles ABC are complementary. And that is because their
measures add up to 90 degrees. So the measure of angle DBA
plus the measure of angle ABC is equal to 90 degrees. They form a right angle
when you add them up. And just as another
point of terminology that's kind of related
to right angles, when a right angle is
formed, the two rays that form the right angle
or the two lines that form that right angle or
the two line segments that form that right angle
are called perpendicular. So because we know that measure
of angle DBC is 90 degrees or that angle DBC is a
right angle, this tells us, we know that the
line segment DB is perpendicular to
line segment BC. Or we could even say ray BD
is-- instead of using the word perpendicular, there's
sometimes this symbol right here, which really
just shows two perpendicular lines-- perpendicular to BC. So all of these are
true statements here. And these come out of
the fact that the angle formed between DB and BC,
that is a 90-degree angle. Now, we have other words
when our two angles add up to other things. So let's say, for example,
I have one angle over here. Let me put some letters
here so we can specify it. So let's say this
is X, Y, and Z. And let's say that the
measure of angle XYZ is equal to 60 degrees. And let's say that you
have another angle that looks like this. And I'll call this,
let's say, maybe MNO. And let's say that the measure
of angle MNO is 120 degrees. So if you were to add the two
measures of these-- so let me write this down. The measure of angle MNO plus
the measure of angle XYZ, this is going to be equal to 120
degrees plus 60 degrees, which is equal to 180 degrees. So if you add these
two things up, you essentially are able to go
all halfway around the circle. Or you could go throughout
the entire half circle or semicircle for a protractor. And when you have two angles
that add up to 180 degrees, we call them supplementary. I know it's a little hard
to remember sometimes. 90 degrees is complementary. They're just
complementing each other. And then if you add
up to 180 degrees, you have supplementary. You have supplementary angles. And if you have two
supplementary angles that are adjacent so that they
share a common side-- so let me draw that over here. So let's say you have one
angle that looks like this. And that you have another angle. So let me put some
letters here again. And I'll start reusing letters. So let's say that this is ABC. And you have another angle
that looks like this. I already used C. Once again, let's say
that this is 50 degrees. And let's say that this right
over here is 130 degrees. Clearly, angle DBA plus angle
ABC, if you add them together, you get 130 degrees plus 50
degrees, which is 180 degrees. So they are supplementary. So let me write that down. Angle DBA and angle
ABC are supplementary. They add up to 180 degrees. But they are also
adjacent angles. And because they're
supplementary and they're adjacent, if you
look at the broader angle, the angle used from the sides
that they don't have in common. If you look at
angle DBC, this is going to be essentially
a straight line, which we can call a straight angle. So I introduced you to
a bunch of words here. And now I think we have
all of the tools we need to start doing
some interesting proofs. And just to review here,
we talked about any angles that add up to 90 degrees are
considered to be complementary. This is adding up to 90 degrees. If they happen to be adjacent,
then the two outside sides will form a right angle. When you have a right angle,
the two sides of a right angle are considered to
be perpendicular. And then if you have two angles
that add up to 180 degrees, they're considered
supplementary. And then if they
happen to be adjacent, they will form a straight angle. Or another way, if you said,
if you have a straight angle and you have one of the
angles, the other angle is going to be
supplementary to it. They're going to add
up to 180 degrees. So I'll leave you there.