Introduction to Angles

Introduction to Angles


Bam! Mr. Tarrou. In this video, we are going to do a nice thorough
introduction of the basic geometric structure or figure of an angle. And through this discussion, we are going
to be talking about the structure of an angle, which is what we’re going to start here. We are going to be talking about how to name
angles off of their size, that’s here. We’re going to be talking about the protractor
postulate, and as well as I can on the board, give you an idea of how to use a protractor
and read measurements from it. We’re going to be talking about the angle
addition postulate, the idea or define what congruent angles are, define adjacent angles,
we have an example of adjacent angles here, even though I’m not going to reference the
actual definition of that till near the end of the video, and define what an angle bisector
is. We are not going to be constructing that using
a compass, just defining that concept for you. So let’s get started, we have a lot to cover. Angles. Well angles, and I’ll be using this sort
of, well, what looks like an angle with a short of arc through the corner of it, that’s
the way I draw my symbol for angles instead of writing the word out, because if I don’t
include this arc, like some text books don’t, I end up writing what essentially looks like
an L, and it can be confusing, so this is going to be the math symbol we use for angles. So angles are formed by two rays that have
the same endpoint. Now remember a ray is not a complete line,
lines extend in both lines forever, it’s not a segment, that stops at, you know, it
has endpoints, two endpoints, whereas a ray only have one endpoint. You can almost think of it as the initial
point, where the ray emits, or originates from. Like say a ray of light. Well it starts with the sun, so that sun would
be the initial source of the ray emanating from it. So, but we’re calling it endpoint. So, these rays have a common endpoint, I’d
probably say an initial point as well because that is vocab. used in some of my other math books from other
math classes, so just kind of interchanging that. So we have point B, we have an initial point,
or an endpoint of B, whatever you want to call it, and then the ray extends in one direction
from there, and each of those two rays form the two sides of your angle. That common point for the rays, the common
endpoint, the common initial point, we’re going to identify that sharp bend there as
the vertex. And then we have a little one marked in here
in the corner of the angle. If you have a, especially if they’re single
digit numbers, there probably, the diameter, was probably giving you an alternative way
of talking about or naming that angle, if you have a little circle in the upper right
hand corner of that number though, that’s indicating the number of degrees that that
angle is open. Angles you know start kind of closed if you
will, and then start to open it up. That amount of opening is that number of degrees
that the angle is sort of encompassing, or has been opened up. So we have two rays, we have a vertex, those
are making up the size. Let’s talk about how to name an angle. Well in geometry, very very basic diagrams
like this, you have many options of how you would like to name that angle. We can use three letters, and most of the
time we name angles in geometry because the diagrams that we are going to use are going
to be a bit more complicated than this, you will need three letters to identify an angle. You’re going to have a point on one side
of the angle, or on one ray, a point on the other ray or side, and the center letter,
or variable, or point, it has to be the vertex. So we can call this angle ABC, keeping the
vertex as the second letter, or you can go in the reverse order, that’s okay, we can
call this CBA, either one of those would be acceptable names for this angle. What would be incorrect would be to name this
as angle CAB, because once you get used to identifying angles, or naming them, you’ll
understand that center letter does need to be the vertex. When you say angle CAB, well that would be
a completely different angle than what I have diagrammed up here on the board. So, that would be incorrect. Keep that vertex as your second letter. If you only have on angle, emanating from
a common point, or a point, if you only have that vertex with one angle coming off of it,
you can name an angle with just simply that vertex, the variable of that vertex, we can
call this angle B, or we can use that inside number, and just call it angle 1. We got another diagram here, these are adjacent
angles, we are going to define what adjacent angles mean in a few minutes, but we’re
still talking about how you name angles. When you have a vertex with, how many angles
do see leaving from this vertex of H, by the way. Do you see two? Or do you see three? Either way that’s a problem with using this
very simple one letter idea of naming the angle. So you actually have three angles coming off
of this vertex of H. You have angle DHF, which I don’t have written up here in my notes,
but sort of like the overall very large angle. And then this, you could think of this angle
of DHF as being split up into two equal parts- DHE, so we have angle DHE, you can reverse
the letters as well, just keep H in the middle. Or, because I have that one in the corner
of the angle, I can call it angle 1 as well. And then we have angle EHF or angle 2, because
I have that 2 marked in my diagram. What would be incorrect, probably should’ve
done this in orange, is because I have a more complex diagram now, I cannot use this simple
idea of naming the angles, say angle H, because well, angle H, what do you mean? The big one? Or this top one up here? The bottom one down here? I don’t know. So, this is now going to be an appropriate
name for that diagram, which is just a little bit more complex than my original one. That simple version is no longer any good. And again, HEF. What’s the problem with this name? E, which is the center letter, for variable
in my name, is not a vertex, that would also be incorrect, and that’s why. Okay. Acute angles. Acute angles measure any kind of angle that
is less or between 0 and 90 degrees. Now what is 90 degrees? I don’t have a protractor in my hand, but
I do have a piece of paper. The corner of your paper, is a 90-degree angle,
this is square, well the entire sheet of paper is a rectangle, but we would say that corner
is square. That corner is a right angle. So, when you’re looking at your textbook,
or a test that you’re doing, or your homework, whatever, you can actually just use the corner
of your paper, and, well let’s just do that now. If I take the corner of my paper, and I put
the corner of my paper on vertex B, do you see how the other side goes underneath the
sheet of paper? Well that means that angle ABC, you know,
opens up less than 90 degrees, so this is an acute angle, it’s between 0 and 90 degrees. An angle that is zero degrees is just shot. And then it opens up and it opens up until
it gets to about 90 degrees, and then it will continue to open up and so on. If you have an angle that’s exactly 90 degrees,
that’s going to be called a right angle. We are going to be talking about right angles
a lot. Cause’ say if you’re building a house,
you want all the walls to have right angles, so it’s not leaning over and have the tendency
to fall over. Right angles are going to help us create right
tringles. And right triangles are a massive part of
the study of an upper level math, called trigonometry, which is extremely important for again building
and navigation, it has tons of applications. You might go in college, and have a semester
course called trigonometry, or a high school course called trigonometry, and it is almost
completely the study of right triangles that are made up of course that one main angle
of a right angle. Obtuse angles measure between 90 and 180 degrees. So, we have 0, we have 90, anything in between
is acute, once you get past 90 between 90 and 180 degrees is called an obtuse angle. Over here, a straight angle, if I take my
arms and just kind of, you know, open them up completely, and you think of my arms as
the size of an angle, if that’s going straight, if it’s 180 degrees, just a straight shot,
like a straight line, if you’re angle looks like a line, then it’s a straight angle. Protractor postulate. On line AB, we’re going to do a diagram
to go along with this. But, on line AB, in a given plane, choose
any point O between A and B. Consider OA and OB and all rays that can be drawn from O on
one side of AB. These rays can be compared with real numbers
from 0 to 180 degrees, in such a way that OA is paired with this any value between,
with 0 and OB with 180 degrees, if OP is paired with X and OQ, ray OP is paired with x and
OQ with y. Ray OQ with y, then measure of angle POQ is
equal to the absolute x minus y. Makes total sense right? Maybe I should have stopped and included the
diagram before I started reading it, because this all sounds like a bunch of gobbly-goop,
that’s why so many people have trouble reading text books, but it does mean something. SO let’s take a pause, erase the left hand
side of the board, give you a diagram, and re-visit this, and make it make sense. So, here we have our protractor drawn. That’s this yellow semi-circle type diagram
here. When you buy these at the store, a protractor
has a straight ruler on the bottom, and there’s usually a like little hole there, where you’re
going to line up with this point O, and I’ll describe it here in a second. And then sort of like, a half circle, a semi-circle,
that we’re going to use to measure angles that have measurements between 0 and 180 degrees. So, again let’s talk with that math language
that’s supposed to tell us how to use a protractor. This postulate: on line AB, so that’s what
I have here, again a line because there’s double arrows here. In a given plane, choose any point O between
A and B, so mark this green dot a point between point A and B, or on this line AB. Between A and B. Consider OA, ray OA, so the
one directional mark here, consider ray OA and ray OB and all other rays that can be
drawn from O, on one side of line AB, we’re going to be measuring angles, and angles are
made up of two rays. That’s why they’re making this point here
in the definition, that we now have taken this line, and created sort of you know two
rays from it, originating from point O. So, we can use that definition of an angle
with our protractor postulate. And we’re only going to be considering all
other rays, and I have 3 others drawn here, I have ray OC, OD, and OE, all drawn on that
same side of line AB. Because this protractor is only going to allow
us to measure angles between 0 and 180, so they all need to be on one side. These rays can be paired up with real numbers
from 0 degrees to 180, so the protractor postulate, you know, a protractor allows you to measure
the number of degrees that an angle opens up. In such a way that OA is paired with O degrees,
and OB is paired up with 180 degrees, so basically what you’re going to do is you’ll have
an angle drawn, so I can kind of cover all this up and maybe just focus on this ray OA,
and OC, so you would put the protractor down on that angle, and you put the protractor
so that the hole on the protractor is going to be over point O, the vertex, and you’re
going to line up ray OA so that that sign of the angle goes through, there’s going
to be a whole bunch of little numbers as you go around these arcs, and you want OA to line
up exactly with vertex. Here at the hole on the protractor, and then
going through 0 degrees, and then as you see the angle, let’s see, and AOC, sort of open
up, you’re going to look at what tick mark, what angle degree measurement the other ray
of the angle goes through, So, we have one side going through OA, going through zero
degrees. You look at the other ray that makes up the
angle and you say, okay it’s lining on top of 130, and I’d also notice that it’s
lining up with 150 degrees, a lot of protractors will have numbers going both this way and
this way if you have one of those. You want to make sure that you know, AOC is
clearly acute, it’s smaller than the corner of a piece of paper, and so you want to use
that smaller number. It’s opening up 30 degrees, and I’ve actually
just answered one of my questions here. The measure of angle AOC is equal to 30 degrees,
and we can just take that number right off of the protractor because the other ray making
up that angle goes through 0 degrees. If OP is, and I don’t have a P up here,
but we’re going to just use different letters, if OP is paired with X, and OQ with Y, then
the measure of angle POQ is equal to the absolute value of X-1. What the heck does that mean? Well back with our protractor, if you’re
able to put one ray right through zero, then you can look at the other ray and just identify
the angle mark, and go okay that’s 30 degrees, and I just wrote 30. But if we look at the measure of angle COD,
okay, so COD, I’ve got kind of a complex here with a bunch of angles going on. Or emanating from the same vertex, and I don’t
want to just move the protractor all the time, and you can if you’re just doing simple
numerical measurements, but were going to ultimately move into doing some problems that
involve a little bit of algebra, and with those more complex questions, you need to
understand how to find measurements of angles without constantly moving that protractor
around. And so, let’s get to it. We have COD. So, we’re going to have this 30 and this
45. So, find what the measurement of COD, what
I’m going to do is take the absolute value, and that is to prevent us from having a negative
angle measurement. Think of a tape measure, you know that sort
of usually metal or plastic box, and you pull the tape out, you can only measure positively. That tape measure doesn’t start whipping
inside and somehow come up with some magical negative measurement. If you’re measuring height, weight, number
of degrees that an angle opens up, there’s only positive measurements, and the absolute
value is going to make sure of that. SO we’re going to take the 30 and the 45,
that would represent sort of our X and Y that’s in part B of this postulate, and subtract
them. Except I wrote 40. Well, 30 minus 45 is negative 15. Well if I had done 45 minus 30, of course
I would’ve gotten that positive angle measurement that we needed. But just in case we do the subtraction in
such way that gives us a negative number, the absolute value makes sure we say, okay
let’s not forget measurements must be positive, and the measure of Angle COD or DOC is 15
degrees, and such, therefore, what is the measurement of angle DOB? Starting here, D is going through the, I’ll
see this 45-degree mark on my protractor, and I’ve got another angle going through
again the vertex here of O going through 120. What is that again? Let’s see. It would be the absolute value of 120 minus
45. Well 120 minus 40 is 80. And 80 minus 5, doing that two steps in my
head, is going to be equal to 75 degrees. So, the final answer is 75. So that is how you, with a diagram now, understand
what the protractor postulate is actually trying to say. And a little bit of how, you know, you use
that protractor when you get one in your hands and you’re working with your homework. Let’s get to the next topic which is going
to be, I don’t know, we’ll figure it out when we come back. Let’s talk about the angle addition postulate. If point B lies in the interior of angle AOC,
then the measure of angle AOB, so we have AOB, keeping that vertex as the enter variable,
plus the measure of angle BOC, BOC, then the sum of those two parts, and it would kind
of make sense, is going to equal the measure of the entire angle AOC. So, AOB, this bottom section plus, BOC, the
measure of angle BOC, this top angle, when you add these two together, you should get
the measure of the entire, sort of, these two parts are part of this outside angle of
AOC, and I can see my face getting a little orange, and a little dark with the clouds
coming in, so let me fix that. Hopefully that’s a little better. And that would make sense. Right? If an angle opens up 10 degrees, and opens
up another 30 degrees, my hand motions are way to big there, but if you open up 10 and
40 degrees, you open up a total of 50 degrees right? It kind of makes sense. Here’s what it’s going to look like with
a little bit of algebra. Got to keep those skills fresh from possibly
last year, depending on where you’re going to school at. But that’s always important, we’re still
going to be mixing up some algebra with our geometry. So, if the measure of the angle AOC, the entire
angle around the outside, if that measurement equals 74 degrees, then let’s go ahead and
solve for X. So, this entire angle around the outside here
has a measurement of 74 degrees, and we have two unknown parts of 3x minus 10 and x plus
7. Well the angle addition postulate simply says
add them up and make them equal to 74. So, we have 3x minus 10 plus x plus 7 is equal
to 74 degrees. I’m not going to keep writing that little
zero there, until the end, but it is 74 degrees. Actually, nevermind, yes I will. I’m only including these parenthesis here
to do a couple of things. I want to highlight the fact that I’m just
simply adding the measurement of the two angles, and it’s always a good idea to use parenthesis
when you do substitution anyways. Like say if we were subtracting the measurements
of the angles for some reason. We would want to remember to distribute that
negative sign through the parenthesis. Okay, so we’re just doing addition, there’s
nothing in front of these parenthesis, no exponents, no fraction bars, so they are kind
of just there to emphasize the two angles. We’re going to add like terms so we have
3x plus 1x. So, 3x plus 1x, we’re going to add the coefficients,
three and one is equal to four. So, we have 4x and negative 10 plus 7. Negative 10 plus 7 or 7 minus ten is going
to be equal to negative three. And that’s equal to 74 degrees. We’re going to go ahead and add three to
both sides, so we have 4x adding that 3 to both sides, negative three plus three is equal
to zero, and 74 plus 3 is equal to 77. And I’m going to kind of drop the degrees
symbol at this point, because when we get done, I don’t want you to think of x as
being the size of either one of these angles. The measurement of the degrees of either one
of those angles. Because x is going to have to be plugged back
into these expressions to find the measurement of COB, and find the measurement of angle
BOA. I didn’t set this example up to find those
two measurements, but that’s what you would do. We are going to divide both sides by four,
and get x is equal to 77, over four. And if you want to write that in a mixed fraction
form, you’ll do 77 divided by four, and that would give you 19, and 19 times four
is 76, so you would have a remainder of one. So this would become a mixed fraction of 19
and one fourth. Just want to make sure that I’m giving you
the right answer. Now this is just the unknown variable of x. If you do want to or need to because your
directions, find the measurement of each of those two angles, you will need to take x,
and plug it into the expression. Three times 77 over four, minus 10 to give
you the actual angle measurement, the number of degrees. And of course the same for the bottom angle
as well. The second part to the angle addition postulate,
just a special case, you know, in case you happen to read a problem that says that the
angle is straight angle, you’re going to have to in geometry, and as well as any other
math class, really work on knowing your vocab. and learning these postulates, and what they
say, off the top of your head. So, if a problem says that an angle is a straight
angle, you need to know that that means it’s 180 degrees, that it opens up, and has a full
180 degrees, you’ll be kind of looking at a straight line, if you don’t know that,
you’ll have some questions that you won’t be able to do. So, if angle AOC (AOC), see how it looks like
it’s a straight line? Is a straight angle, and B is any point not
on AC, and the really, the ABC thing is really for the definition. These could be identified with any variables
you like, or your textbook may like to use. But then the measure of angle AOB (AOB), plus
the measure of BOC, (BOC), keeping those vertexes in the center there, is equal to 180 degrees. Because a straight angle is an angle whose
opened up a full 180 degrees. Given that AOC is a straight angle, solve
for X. Here, yellow, with these expressions in here,
that’s all your textbook should have to tell you to set up to solve for x. They don’t have to tell you that the angle
is 180 degrees. When they tell you it’s a straight angle. That’s for you to know. So we have, well, we have these angles, were
going to define adjacent angles here in a second, we have these adjacent angles, they
are adding together, coming to together to add up to this full angle of AOC. So, the angle addition postulate says take
this first part, add it with the second part, and make sure that equals 180 degrees. And this will finish just like this problem
did over here. X and 5x is equal to 6x minus 10 is 180. We’re going to add 10 to both sides of the
equation. Whenever you want to move a number, or a variable
for that matter, from one side of an equation to the other, you always want to do the opposite
math operation, so that’s going to be six x is equal to 190, and divide both sides by
six, and we get, let’s see here, oh I’ll just cheat, [chuckling] 63 and a third. Wait a minute, yeah. No this can’t be right. I found a little mistake in my notes there. Okay, so we got to 190 over 6, that reduces
to 95 over three. These both have a divisor of two, and if you
want to make that improper fraction into a mixed fraction, I think I called it a compound
fraction over here, sorry about that. So, if you want convert that to a mixed fraction,
95 divided by three, is 31 point [mumbling]. So, there’s a decimal afterwards. So, 31 times three, is equal to 93. Okay, 31 times 3 is equal to 93, ad 93 from
95, gives you a remainder of 2, so we have thirty-one and two thirds. So, that is the angle addition postulate. Let’s define what congruent angles are,
adjacent angles are, and define an angle bisector, and this video will be done. Hopefully this lecture will help you get your
homework finished. Thank you for watching. Here we go, last screen. Congruent angles have equal measurements,
so we have an angle here drawn, I have just identified it with one letter, and angle DCB. Those angles, if given, angle A is congruent
to angle DCB. Well if that’s the case, then if you’re
told that they’re congruent, then you can say that the measure of angle A is equal to
the measure of angle DCB. Another thing to be aware of, is when you
look at your math diagrams if you have some angles drawn like this, and I can tell you
that these angles are congruent, or with this symbol, I can draw some arcs through here. In geometry, we’re going to use those little
arcs we put through the corners of the angles as equal signs. So, if I do this, in my diagram, I am indicating
that those two angles are equal in measurement, and thus they will also be congruent. Please note the congruent symbol, the equal
sign, we talk about measurements, the M here, measurements being equal. So, it’s not the same to say that two angles
are congruent, it’s not the same as saying that their measurements are equal. This definition goes back and forth, but it’s
not the same notation. Adjacent angles are two angles in a plane
that have a common vertex, and a common side, but not a common interior point. So, I have four diagrams here, that, you know,
could identify or be angles that are adjacent. So, which ones are and are not? Well, with our first one here, we have angle
1, and angle 2. The yellow and blue one. We have a common vertex, and they are of course
sharing this entire side, this entire ray, which I don’t have any letters to name it,
but we do have a common vertex, and a common ray shared between angle 1 and 2. So these are adjacent. In this diagram, we have two angles, angles
one and angle 2 sharing a common vertex, but you can see that angle 1 is made up of these
two yellow rays, and angle 2 is made up of these two blue rays. And nothing else is in common except for the
vertex, so those are not adjacent. We have angle 1 and angle 2 sharing a common
ray, partially, angle 2 shares most of this ray, but not all of it. See they don’t have the same vertex, so
these are not going to be adjacent angles. And, over here, we have a common side, that’s
good, we have a common vertex, being shared between angle one and angle two, now if I
look at here, and I look at here, you might be going well, so let’s just put another
arc through here, I’m just really trying to identify how far angle one opens up, that
it goes from this side to this side, and I’m not trying to identify as those being equal,
so let me change that a little bit. So, we have a common side and a common vertex,
but you can see that angle 2 is within angle one, so they’re sharing some interior point. So, this is also an example of two angles
which are not adjacent. Our last definition, angle bisector, the bisector
of an angle is a ray that divides the angle, ultimately, we’re going to be talking about
here with angle DEF. In the two congruent adjacent angles. So, all I have here though, is from my diagram,
you can assume that all points are co-planer. I’m talking about DHF and E, are all on
the chalkboard, so they’re all co-planer. You can see that ray ED and ray EH and ray
EF all intersect, they also all have the common endpoint, or initial point of E, and you can
see that angle DEH, and angle HEF are adjacent. They share this angle here DEH, and HEF share
that vertex and that common side. So, that they are adjacent. But you cannot assume that these angles are
congruent without some marks in here. So, if this were your diagram in your textbook,
you could not look at that and assume that these two angles are equal. But you can, you’re not assuming that they’re
equal, your concluding, or being given that those angles are equal if you have those little
arcs in the angle measurements. If there’s a bunch of angles, you might
see two arcs or three arcs, but those angles that have the same numbers of these little
arcs in the corners, that’s your textbook telling you that those are equal. So, uh, what I’m want to do with this, well
let’s these here, I want to make sure I don’t miss something that I wanted to include. We have angle DEH, is congruent to HEF, that
means that they’re, let’s see here, angle DEH is congruent to measure angle HEF, that
now of course means their measurements are equal. And let’s see, let’s pick on this angle
here, HEF. The measure of angle HEF and I want to compare
that to the measurement of the entire angle. The outside angle, DEF. Oh. Correct my little angle symbol there. DEF is the entire angle right? And it’s been bisected or cut in half, so
if I just look at HEF, which is half of the entire angle, then I can say that twice the
measurement of this small part, that takes two of these halves to equal the measurement
of the entire angle that was initially bisected or cut in half. I could also say that the measure of HEF,
is equal to one half the measure of the original angle, DEF, that was then bisected by a ray
EH. Well that is the end of my introduction to
angles. And many, many definitions included in the
video. I hope you appreciate the fact that I am wrapping
all of these up, and not making you click through a bunch of small little videos and
give you one, you know, nice, concise, complete, lecture to help you get your homework done. So, now I’m done it’s time for you to
go do your homework! Bam!

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