Angles in Circles Chords Secants Tangents and Arcs

Angles in Circles Chords Secants Tangents and Arcs


BAM! Mr. Tarrou. In this video we are going to work through
three multi-part examples, so really you can argue that it is more than three examples,
but at any rate… Looking at angles that are created when two
lines are intersecting a circle. Now we can have that intersection in the middle
of the circle, or inside of the circle. Not in the middle, that would be the center. It does not have to be the center. Or we can have that interaction point outside
of the circle. So the measure of an angle, here is angle
A,… The measure of the angle formed by two lines
that intersect inside of a circle is half of the measure of the sum of, probably should
have added that, of the sum of the intercepted arcs. So we are going to say… ok… if two lines
intersect in the circle, the intersection point is inside… You can of think of these, remember that a… I can’t really call them chords because a
chord is a segment whose endpoints are on the circle, but these lines are intersecting
the circle twice. So if I were just erase the portion that was
extending beyond the circle, they would be chords. You can think of I have an intersection of
two chords if you will. And this angle is the average of the sum of
these two arcs. Now we are measuring these arcs in degrees
not in length. So here we go. The measure of angle A is equal to 1/2 the
sum of those two arcs created or between those two lines intersecting inside the circle. Well, what happens when the intersection of
these lines are outside of the circle? I can see these colors. Hopefully you can see them in the camera. I have two yellow lines. Both of them intersecting this circle twice,
and the intersection point is on the outside. I could kind of call these chords again, but
they are not segments. They are actual lines, or complete lines. So um… And they extend on actually, you know, on
through this intersection. I just wanted to make a neater picture. I have some… I have an orange angle if you can see that. It is consisted of a line intersecting the
circle twice and a line intersecting the circle only one time. That means that this orange line and it is
also bluish purple if you can see that. Maybe you cannot see that. But this is a tangent line touching this circle
only once. Or, we have an intersection of two lines that
each of those lines are a tangent line. And either way, the intersection point of
those two lines are outside of the circle. If that is the case, and there is three cases
up here if you will, we have the measure of angle A is equal to one-half.. now not the
sum of these arcs… but the difference. Now my textbook shows this with parenthesis. 1/2 of the difference of arc x minus the arc
of y. Your book might have some absolute value symbols
here because you cannot have a negative measurement. And if your textbook uses absolute value symbols,
that simply means that if you happen to accidentally do the small arc minus the larger one and
get a negative difference the absolute values will take care of that and make it positive. Making sure that you end up with a positive
measure for angle A. Ok. So I guess I am just going to read now what
I verbally said. The measure of an angle formed by two lines
that intersect outside of a circle is half the difference of the measures of the intercepted
arcs. Ok. So let’s get to these three examples and see
how all of this works out. Hey! Before we start our first example, I normally
have my external microphone on my camera set on Mono to try and help… I am afraid it might sound too echo’y in this
room, this relatively small room as I shoot these videos. And I started the video off accidentally with
my microphone on stereo. So I don’t normally or ever asked if there
is a preference. If you noticed a sound difference, sound change,
and you like the first part better than the second part… you like the mono better than
the stereo which I think I like better, let me know. Ok, at any rate. Our first example. If the measure of angle… Now I am going to do two examples off of this
one circle. That is why I am defining these variables
here. If the measure of arc AB is equal to 38 degrees
and the measure of arc CD is 74, certainly not drawn to scale, what is the measure of
angle COD. You should never trust the scale of a diagram
anyway. This is the first example so it is going to
be the most straight forward. I have two, well I did not draw the lines
this time, I actually just drew the chords. The section or the part of the lines that
are intersecting inside of the circle. And I want to find the measure of that angle
between these intersected arcs. Ok, well. I just said in the previous screen there was
sometimes we took half of the sum and sometimes we took half of the difference. And maybe I have already forgotten. Well, this is a bit of a stretch but these
two lines are intersecting inside of the circle and it kind of looks like a squashed plus
sign. Maybe that will be your reminder for you to
add those measures of the arcs before you divide by two to find the measure of that
angle. So we are going to add. So the measure of angle COD, that is not Cash
On Delivery, is going to be equal to… Well, actually it is but we are not taking
delivery we are doing some Geometry. This is equal to the average, or half… i
keep saying average because I am adding two numbers and I am dividing by 2. That would be finding the average. If I were adding three numbers and dividing
by three, that would be the average as well. Adding three numbers and dividing by 2, is
just taking half of the sum. We are going to take half of that sum. 38 plus 74. Let’s see here. That is going to be 1/2 of… 70 plus 30 is
100… now we are at 108… and now we are at 112. And half of 112, well let’s see here. 11 divided by 2 is 5, but 5 times 2 is 10. That leaves me a remainder of 1, so I have
12 left over. Half of 12 is equal to…. now 56 degrees. So the measure of angle COD with our first
example is equal to 56 degrees. Ok. BAM! Mr. Tarrou. In this video we are going
to work through three multi-part examples, so really you can argue that it is more than
three examples, but at any rate… Looking at angles that are created when two lines
are intersecting a circle. Now we can have that intersection in the middle of the circle,
or inside of the circle. Not in the middle, that would be the center. It does not have
to be the center. Or we can have that interaction point outside of the circle. So the measure
of an angle, here is angle A,… The measure of the angle formed by two lines that intersect
inside of a circle is half of the measure of the sum of, probably should have added
that, of the sum of the intercepted arcs. So we are going to say… ok… if two lines
intersect in the circle, the intersection point is inside… You can of think of these,
remember that a… I can’t really call them chords because a chord is a segment whose
endpoints are on the circle, but these lines are intersecting the circle twice. So if I
were just erase the portion that was extending beyond the circle, they would be chords. You
can think of I have an intersection of two chords if you will. And this angle is the
average of the sum of these two arcs. Now we are measuring these arcs in degrees not
in length. So here we go. The measure of angle A is equal to 1/2 the sum of those two arcs
created or between those two lines intersecting inside the circle. Well, what happens when
the intersection of these lines are outside of the circle? I can see these colors. Hopefully
you can see them in the camera. I have two yellow lines. Both of them intersecting this
circle twice, and the intersection point is on the outside. I could kind of call these
chords again, but they are not segments. They are actual lines, or complete lines. So um…
And they extend on actually, you know, on through this intersection. I just wanted to
make a neater picture. I have some… I have an orange angle if you can see that. It is
consisted of a line intersecting the circle twice and a line intersecting the circle only
one time. That means that this orange line and it is also bluish purple if you can see
that. Maybe you cannot see that. But this is a tangent line touching this circle only
once. Or, we have an intersection of two lines that each of those lines are a tangent line.
And either way, the intersection point of those two lines are outside of the circle.
If that is the case, and there is three cases up here if you will, we have the measure of
angle A is equal to one-half.. now not the sum of these arcs… but the difference. Now
my textbook shows this with parenthesis. 1/2 of the difference of arc x minus the arc of
y. Your book might have some absolute value symbols here because you cannot have a negative
measurement. And if your textbook uses absolute value symbols, that simply means that if you
happen to accidentally do the small arc minus the larger one and get a negative difference
the absolute values will take care of that and make it positive. Making sure that you
end up with a positive measure for angle A. Ok. So I guess I am just going to read now
what I verbally said. The measure of an angle formed by two lines that intersect outside
of a circle is half the difference of the measures of the intercepted arcs. Ok. So let’s
get to these three examples and see how all of this works out. Hey! Before we start our
first example, I normally have my external microphone on my camera set on Mono to try
and help… I am afraid it might sound too echo’y in this room, this relatively small
room as I shoot these videos. And I started the video off accidentally with my microphone
on stereo. So I don’t normally or ever asked if there is a preference. If you noticed a
sound difference, sound change, and you like the first part better than the second part…
you like the mono better than the stereo which I think I like better, let me know. Ok, at
any rate. Our first example. If the measure of angle… Now I am going to do two examples
off of this one circle. That is why I am defining these variables here. If the measure of arc
AB is equal to 38 degrees and the measure of arc CD is 74, certainly not drawn to scale,
what is the measure of angle COD. You should never trust the scale of a diagram anyway.
This is the first example so it is going to be the most straight forward. I have two,
well I did not draw the lines this time, I actually just drew the chords. The section
or the part of the lines that are intersecting inside of the circle. And I want to find the
measure of that angle between these intersected arcs. Ok, well. I just said in the previous
screen there was sometimes we took half of the sum and sometimes we took half of the
difference. And maybe I have already forgotten. Well, this is a bit of a stretch but these
two lines are intersecting inside of the circle and it kind of looks like a squashed plus
sign. Maybe that will be your reminder for you to add those measures of the arcs before
you divide by two to find the measure of that angle. So we are going to add. So the measure
of angle COD, that is not Cash On Delivery, is going to be equal to… Well, actually
it is but we are not taking delivery we are doing some Geometry. This is equal to the
average, or half… i keep saying average because I am adding two numbers and I am dividing
by 2. That would be finding the average. If I were adding three numbers and dividing by
three, that would be the average as well. Adding three numbers and dividing by 2, is
just taking half of the sum. We are going to take half of that sum. 38 plus 74. Let’s
see here. That is going to be 1/2 of… 70 plus 30 is 100… now we are at 108… and
now we are at 112. And half of 112, well let’s see here. 11 divided by 2 is 5, but 5 times
2 is 10. That leaves me a remainder of 1, so I have 12 left over. Half of 12 is equal
to…. now 56 degrees. So the measure of angle COD with our first example is equal to 56
degrees. Ok. I am going to erase these numbers because this second example here, part B I
want to use the same diagram but… I was too lazy to make another. I am going to erase
those numbers and I am going to start over like it is a brand new problem. The measure
of angle AOB is equal to 46 degrees. Now we just said that COD was 56, and vertical angles
are on opposite sides of an intersection of two lines. So if I was extending the same
question angle AOB would be 56 degrees. So remember we are starting a brand new problem
Probably should have said this was example 2 instead of part B. So the measure of angle
AOB is equal to 46 degree. We have the measure of arc AB is equal to 10. Except that would
have to be 10 degrees. If I just leave that without the degree mark you might think it
a length. And I want to find the measure of arc CD. Ok. Well, I still have an intersection
of two lines.. or chords… inside of our circle. We are still going to say that the
measure of this angle AOB is 1/2 the sum of the two arcs. It is going to be 1/2 the sum
of arc AB and arc CD. Ok. So I am writing this out here because this second question
is a little bit different. The first one was very straight forward. I gave you the two
arcs and I asked for the angle. So we just simply added those two arcs, divided by 2,
boom there is our answer. Well now this one gives us the angle where these two lines are
intersecting and it is asking for the measure of the degree of one of the arcs. Well, if
you have any doubt of how to solve a problem the formula tells you what to do. It tells
you what it needs and where it belongs. If you just plug those numbers in where they
belong, and hopefully with your algebra skills you will know how to finish the problem. So
I am writing the formula out and we are told that angle AOB is 46 degrees. I am going to
take this out, and put in the 46. The measure of arc AB is 10 degrees. So we are going to
take that out and put in 10 degrees. And ok. Well, I am looking for the measure of arc
CD. How am I going to figure it out. One way of doing this is to take the 1/2 and distribute
it through the parenthesis which I think I am going to do. What you could also do is
recognize that this entire parenthesis is being locked together with the parenthesis,
and this one this is being multiplied by a half. So you could multiply both sides by
2 to undo that division of 2. So either way it is going to be relatively the same number
of steps. I am going to go ahead and take the 1/2 and distribute it through the parenthesis.
So we have 46… 1/2 of 10 is equal to 5… and then we get plus 1/2 of the measure of
arc CD. We are going to start solving this equation. That means we need to add or subtract
away from the variable first. We are going to subtract both sides by five. And apparently
start to run out of room. We got 41 is equal to 1/2 the measure of arc CD. We will have
enough room to finish this. Again I have that multiplication, or excuse me, I have a fraction.
Now fractions represent division. So we can get rid of this fraction in two ways. Now
with the numerator being 1 either way is going to seem like they are equally… or equal
number of steps. But if you have fractions and struggle with them you can get rid of
them in two steps. You can undo that division of 2 by multiplying by 2, which is what it
is going to seem like I am doing. But if this numerator was something other 1 and I just
multiplied both sides by the denominator… which I could… I would still have to get
rid of that numerator. Like if this were 3/2 and I multiplied both sides by 2, I would
have 82 is equal to 3 times the measure of arc CD and I would then have to divide both
sides by 3. Or, if you are more comfortable with fractions, and again this has a numerator
of 1 so it is not going to seem like it is any different, but if you have a fraction
you can get rid of it in one step instead of two by multiplying both sides by the reciprocal.
So I am going to take the 1/2 and flip it, and multiply both sides by 2/1. And when I
do that, that multiplication of 2… or that multiplication of 2/1 is going to cancel out
with that fraction of 1/2. It just leaves me with 2 times 41 is equal to 82. And that
is equal to the measure of arc CD. And let me just check my own work. Excellent. Ok,
over here now we have a couple of lines that are intersecting the circle but intersecting
themselves outside of the circle. So that means to find the first question here… Now
this is a two part question, but I don’t have if something something equals this solve for
this.. and if something something equals this… So these two parts are coming off the same
diagram. I already have the numbers labelled in there. So the measure of angle x… well
it created by two lines that are involving the circle. Now these two lines both happen
to be tangent lines. But that does not matter. There are involving the circle and they are
intersecting outside the circle. So the measure of angle x is going to be equal to the difference
of the two arcs divided by 2. So the measure of angle x is going to be equal to 1/2 the
difference of the two arcs. Now I want to take the big minus the small so I don’t have
to worry about absolute values. um. Ok, so the intersection points are right here…
the tangency points. And that means I have an arc AD which has the measure of 100 degrees.
Then I have a measurement of an arc that goes more than half way around the circle. So I
can call this arc AFD or AED. Of course I can go in the opposite direction. But the
point is I need to know the measure of this entire arc. That is simple. A full rotation
around a circle, I don’t know why I just made my hand go around twice, but a full rotation
of the circle is 360 degrees. And these two tangency points, if I am only concerned about
these two points around the circle, then I have only got two arcs. I have got one arc
that is 100 degrees, and I have another arc… I mean it is broken up into three parts but
I want the measure of the whole thing, I have another arc that has to add up with 100 to
make 360. Of course that is 260. So we have the measure of angle x is equal to 1/2 times
260 minus 100. That is going to be equal to 1/2 of 160 which is equal to 80 degrees. So
the measure of angle x is equal to 80 degrees. So the measure of angle x is equal to 80.
Now this one is going to make us find some information. Now it is umm… it is related
to this answer, but it is not the answer. I want to find the measure of arc FE. Well
let’s get some orange. So I want to find the measure of this arc. Well, this arc is created
by these two lines, or chords since they do not run forever…. the are ending on the
circle, but these two chords are intersecting inside the circle. Ok, so the measure of arc
FE. Well, on the other side of these lines that are intersecting I have arc AD. If I
don’t figure out based on other information in the diagram some more information with
these two arcs I am not going to be able to find this answer. I need to know something
about angle C, or one of these two angles. They are vertical angles, they are going to
be equal…FCE or ACD to be able to find the measure of arc FE. Well that is right here.
This is 60 degrees. Now that is not going to be using any formula involving or working
towards directly finding the measure of arc FE. I need to know this angle. Now you see
these two angles here, right. We have angle ACF and FCE, these are adjacent angles forming
a straight line which means they are a linear pair. That means they add up to 180 degrees,
or in other words we call that supplementary. So what angle plus 60 is equal to 180? Well
of course that is 120 degrees. Ok, so what we have here is the measure of angle ACD…
or FCE which is 120 is equal to 1/2 the difference of the two arcs that are… Excuse me, not
the difference. It is inside the circle. The plus sign sort of squished over. The sum….
That 120 is going to be equal to 1/2 the sum of these two arcs. So we have 1/2 times…
1/2 of the sum of the measure of arc FE plus 100 degrees. I think this time I am going
to maybe multiply both sides by the reciprocal of 1/2 just to show you another way of showing
the work. We are going to pick up some yellow chalk I guess. Multiply both sides by 2. That
is going to give us 2 times 120 is 240 equals the measure of FE plus 100. We are going to
subtract both sides by 100. We get our answer, the measure of arc FE is equal to 140 degrees.
And yet again, we have a diagram that is not drawn to scale. But, I did check my work before
making the videos and these notes match my notes so I am pretty confident we are good
to go. Don’t trust your diagrams. They are very rarely drawn to scale. One more example.
Whoo! Before I go onto the next screen I just kind of want you to keep your mind open to
perspective and how things should look. If I go from this point and do my best to draw
a diagonal, or excuse me, a diameter through this circle… I can maybe… somewhere around
there. You can see that this original arc that I marked off as being 100 degrees, well
if this is the diameter 100 degrees would be just a little bit more than 90 degrees.
So basically if I were to redrawn this and make it… you know.. really very good drawing
to scale I would have to take this intersection point out here from the circle and push it
much much closer to the circle. Maybe have a diagram that looks something more like this.
My tangent line is intersecting at B. I just wanted to bring that up. Now, last scene.
Our last example has four parts. Ok. I don’t care which one of these you find first or
that we find first. Fe Fi Fo, I don’t know. Let’s do one. Ok, Now we have to know that
this segment here of AE is a diameter otherwise we won’t know enough information to be able
to finish this problem. Any ideas? [laughing] How about we find the measure of arc BC first.
This is a diameter and that means that on each side of this diameter we have a semicircle.
So this arc of AE and this arc of AE, or ABE and AGE, they have to be a 180 degrees each.
And this 180 degrees is making up three parts. One part is 50, one part is 70, and there
is only one part left so it has to… the measure of arc BC has to make the three add
up to 180. So the 180 degrees is going to be equal to 50 plus 70 plus the measure of
arc BC. So we are going to take these two and add them together and get 180 is equal
to 120 plus the measure of arc BC. Subtract both sides by 120. That is two examples where
I put my variable on the right hand side. I don’t know exactly why I keep doing that,
but of course that is probably good for you. A lot of textbooks put the variables on the
left hand side and you just get used to moving all of the numbers to the right. When you
are solving an equation of course you just want to move the known values away from the
variable. If the variable is on the right, everything goes to the left. Anyway that is
equal to 60 degrees. That is the measure of arc BC. I am going to go ahead and take this
3 out and just put in here our answer of 60 degrees. Ok. Let’s go ahead and find the measure
of arc GE. Now arc GE is created by a couple of lines intersecting the circle. Now there
is a Secant line intersecting the circle in two places and then we have a line intersecting
the circle at only place. So this would be a like an extended chord and a tangent line.
And the measure of this angle formed outside of the circle is equal to 1/2 the difference…
1/2 the difference of the two arcs which it creates. So the measure of angle F is equal
to 48 degrees is going to be equal to 1/2… Now let’s put the larger arc first again so
we don’t have to worry about negatvie measurements. And we have got … let’s see here. We have
got this arc of GE and the other arc that is inside this angle , it is kind of tempting
to look at this quickly and just go all the way around the circle. But the other arc that
is created by this angle is right here. Now that arc is 180 degrees. You don’t want to
include this arc AG because it is outside the angle these two lines that are helping
to make this angle. So 40 degrees is equal to 1/2 the difference of 180 degrees again,
that is why we needed to know that was the diameter AE, so 180 minus… well that is
what we are looking for the measure of arc GE. Some say that they bring good things to
life. We are going to… look my variable is on the right side again. We are going to
multiply both sides by 2 to undo this multiplication of 1/2. So… we are going to go ahead and
multiply both sides by 2. We are going to get 96 is equal to 180 minus the measure of
arc GE. We are going to subtract both sides by 180. Now you might be getting concerned
because we are looking for a measure of an arc and the measure right now is coming out
to be negative. But we have a negative here that we are going to divide to both sides
to take care of that. Ok, 96 minus 180. Let’s do this in reverse and just know that the
difference is going to be negative because 180 is larger than… well the larger absolute
value comes from the negative number. 180 minus 90 is 80… and then 80 minus 6 is 74.
It will be negative 74. We are going to divide both sides by negative one to get it away
from the variable that we are looking for. And we get 74 degrees is equal to the measure
of arc GE. So question one is equal to 74 degrees. Ok. Let me just make sure that this
is what I got in my notes as well. hmmm.. [haha] 180 minus 90, what was that again?
I said that was 80, but 180 minus 90 is 90… now minus 4 is 84. Some of you are probably
already commenting, HEY, you can’t subtract. It is supposed to be 84, not 74. I caught
it. Ok. That is why it is always good.. I am talking and trying to do math at the same
time. That does not always work. Make sure that when you are doing your homework you
are staying focused on it as well. Ok, so let’s find the measure of arc AG. Now that
we know that this arc GE is 84 degrees, again this is not drawn to scale, and again this
is a diameter AE. So that means that these two parts also have to add up to 180 degrees.
So the measure of arc AG is 180 minus 84. Let’s try and do this right this time. 180
minus 80 is 100… and 100 minus 4 is 96. Ok. Let’s just check one more time. AG, yes
is 96 degrees. Well the only thing left is the measure of angle 4, or the measure of
angle ADE. Ok, well now angle 4 is from the intersection of two secants inside the circle,
kind of.. sort of look at this as a warped plus sign. Remember the measure of an angle
formed by two secants intersecting a circle is 1/2 the sum of the two arcs. So the measure
of angle ADE, or you can say angle BDC, is equal to 1/2… ok… What arcs did this intersection
make? There is an arc over here of.. Hope you can see the green… there is an arc here
of 60 degrees. There is another arc here, ok it is ending at the diameter so we have
that big arc of 180 degrees again. So 1/2 60 plus 180. That is going to be, OH LOOK
my variable is on the left finally for a change… That is going to be equal to 1/2 of 180 plus
60 which is going to be equal to 240. And 1/2 of 240 is 120 degrees. So, I am Mr. Tarrou.
Time for you to BAM… GO DO YOUR HOMEWORK!

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