Angles Related to a Circle - Mathematics, Class 9 Video Lecture

hello and welcome to Otton math in this
edition of Otton math we're going to
talk about the inscribed circles or
inscribed angles we're to talk about
tangent tangent angles secant secant
angles angles inscribed in a circle cord
cord angles as we apply them to practice
problems and angles related to a circle
all right so let's get started first
problem number 32 from the book a
quadrilateral ABCD is inscribed in a
circle its diagonals intersect at X if
arc AV is equal to 100 degrees BC is
equal to 50 and ad is congruent to BD
and find the measure of angle D XC so
part of the problem or part of what you
need to accomplish with this particular
problem is to draw the figure because
it's not given to you so I'm going to
ask you to take a moment to do that if
you already haven't while I pause
all right so I've drawn out the figure
for you and so you have a circle I have
two chords AC and BD they intersect at a
point X here and what I need to do is I
need to identify what the measure of C X
D is so in order to do that I need to
figure out what arc length or arc
measure CD is so we're going to work
towards that end understanding that I
want to get the equation that X is equal
to 1/2 the sum of the measure of Arc CD
plus the measure of Arc a B and of
course you know we already have the
measure of Arc a/b it's 100 100 degrees
right now we're given that BD and AD are
congruent also that arc B a is 100
degrees so I know that the angle measure
ADB is going to be equal to 50 degrees
so let's mark that up in the diagram
and because I know that the sum of the
measures of all the angles in a triangle
equal 180 degrees and I know that angle
abd and DA B are congruent
I can then conclude that angle DBA and
angle da B are both 65 degrees right if
that's the case then I know that the
measure of Arc BC D is going to be equal
to 130 degrees 130 so let me write that
here the measure of Arc B CD is going to
be equal to 2 times the measure of the
inscribed angle B ad so that is equal to
130 degrees I know that the measure of C
B is 50 degrees so I subtract C B to get
CD and I end up with the measure of Arc
CD as 80 degrees right now that I have
the measure of arc CD I can figure out
what the measure of angle C X D is I
just fill in the values here one half
the measure of CD which is 80 degrees
plus a B which is 100 degrees and I end
up with angle C X D is equal to 90
degrees okay moving on to the next
problem I'm given a circle and two
secant secant or a secant secant angle
with two secant's wz is congruent to XY
arc W X Y is 200 degrees so we're going
to go ahead and mark that up
so let me mark this up here in a second
diagram so W XY is equal to 200 degrees
they need to find the measure of angle P
here all right well I know that the
measure of angle P is going to be one
half the difference of W X and Z Y so
let's write what we're trying to
accomplish here
okay so I need to find out what x and y
are well I know that X W X Y this
measure of this arc is going to be 200
and I'm gonna identify arc W X as X and
arc Z Y as Y so I know that this arc
length or arc measure here is going to
be 200 minus X well I know that this
angle here and this angle here X and W
are congruent I have two cords that are
going to be congruent W Z and X Y so I
know that there are intercepted arcs are
going to be congruent so I can say that
X Y arc X Y is congruent to W Z so if X
Y is 200 minus X I can say that W Z is
also 200 minus X so now I know that the
sum of all of the measures of all of the
arcs is going to be 360 degrees so I can
add them all together I have 200 minus X
plus 200 minus X plus X plus y is equal
to 360 degrees and I'm going to simplify
this I have minus X minus X plus X so I
end up with minus X I have 400 now minus
X plus y equal to 360 degrees so I'm
going to subtract 360 from both sides
and I'm going to add X and subtract Y
from both sides and I get 40 is equal to
X minus y well that's really what I want
right W X is equal to X
and Z Y is equal to Y so I know the
difference between the two is going to
be 40 degrees so my answer is going to
be half of that or 20 so 1/2 of X minus
y or one-half of 40 is equal to 20
degrees all right moving on to the next
problem
a secant did a tangent to a circle
intersect to form an angle of 38 degrees
so it's given here the measures of the
arcs intercepted by this angle around a
ratio of 2 to 1 find the measure of the
third arc all right so we know that for
this equation we have a tangent and the
secant I know that 38 degrees is going
to be equal to 1/2 the difference of the
two arcs so 2x minus X so I know that 38
is equal to 1/2 of X so X ends up being
76 degrees right if X is 76 degrees then
this value here is 152 degrees this
measure of this arc here is 76 degrees
if I add these two together I get two
hundred twenty-eight degrees and then I
subtract that from 360 degrees and I end
up with 132 degrees as my answer
okay moving on so the measure of this
third arc which we can say is our let's
just say it's arc a B is going to be 132
degrees okay two more problems left
number 35 given circles P and Q are
internally tangent at T diameter and s
here the diameter NS of circle Q is
tangent to P at a and 80 so here is 80
so diameter NS is tangent to circle P at
a and 80 the measure of Arc mr here is
42 degrees TM passes through a so TN
pests ray we're going to find the
measure of Arc NM all right so I know
that in inscribed angles half the
measure of its intercepted arc I know
that arc mr is 42 degrees so I can say
that angle a TQ is going to be equal to
21 degrees so I'm gonna write that here
angle 80 Q is equal to 21 degrees well I
know that the measure of the intercepted
arc akin is going to be twice that of
the inscribed angle so I can say that
I'm going to say that this is B here I'm
going to say the measure of Arc a B is
going to be the same as arc mr which is
equal to 42 degrees well I know that tpq
is a diameter so I know that T a q oh
sorry ta be the measure of ta be
we'll be 180 degrees because tpq is a
diameter our TPB is a diameter and now
ta be is equal to 180 degrees so the
measure of ta is equal to 138 degrees so
now the measure of Arc ta equal to 138
degrees now I can find out what the
measure of this angle here is a cutie
the measure of a cutie angle a cutie is
going to be equal to one half of the
difference of 138 minus 42 so let's
write that in there
and this should be okay so a cutie is
going to be one-half of the difference
of Arc ta minus arc a be so 138 minus 42
and this gives us 1/2 of 96 which is
equal to 48 degrees so I know the
measure of a cutie is equal to 48
degrees now I can figure out what the
arc length is of NT and of RS well I
know that the measure of our that angle
n QT or a cutie is a central angle so I
know that the measure of a cutie is
going to be the same as the measure of
NT all right so the measure of NT is
going to be equal to 48 degrees this is
a measure of Arc NT is 48 degrees so
let's write that in here and I know that
I have a diameter and tqr and I know the
diameter cuts off an arc of 180 degrees
so I know that TMR the measure of Arc
TMR is equal to 180 degrees and so I can
figure out the measure of nm by
subtracting NT and M R so let's just
draw that in here I can subtract mr and
NT which will leave me with the balance
of nm and that's what we're looking for
so I have 180 degrees so here is my 180
degree arc and now I'm going to subtract
NT which is 48 degrees and I'm going to
subtract M R which is 42 degrees and I
end up with nm
a measure of nm which is going to end up
being 90 degrees alright on to our last
problem I have two circles intersect at
a and B if the measure of angle ax B so
ax B here is 70 degrees the measure of
Arc CD is 20 and I've written that in
here and the measure of Arc EF here is
160 degrees find the difference between
the measures of Arc a B of the smaller
circle and a B of the larger circle
right so I've written this up for you
now I know that the arc of the larger
circle so this arc in blue here is going
to be one-half of 160 minus a be the
measure of a B so let's back up a second
I'm going to figure out the measure of
Arc a B and the measure of Arc a B is
part of the equation one half of 160
minus this arc length here is equal to
70 degrees so I have two secant secant
angles the measure of this angle here is
equal to the difference of 161 half the
difference of 160 minus a B so I write
in the equation one half of 160 minus a
B and that's going to be equal to 70 and
then I solve for a B alright so a B ends
up being equal to 20 so this value here
of the larger circle ends up being 20
degrees so the measure of the arc a B of
the larger circle is 20 degrees now I
apply the same formula for the smaller
circle and I identify X as the value
that I'm trying to achieve in this case
I want to figure out what this measure
of a B is
so in this case I have instead of two
secant secant angles with an angle
outside of the circle I have to chord
chord angles with the angle inside of
the circle or the vertex inside of the
circle so my equation here is going to
be one one-half seventy is equal to one
half the sum of arc a B which identified
as X plus CD which is identified as 20
degrees so I have 1/2 of X plus 20 is
equal to 70 and I solve for X and I end
up with the measure of Arc a B is equal
to 120 degrees alright that's it for oh
I'm sorry I didn't finish the problem so
I have the measure of the smaller or the
arc for the smaller circle is 120
degrees the measure of the arc for the
larger circle is 20 the difference
between the two is going to end up being
100 degrees all right that's it for odd
math and this addition of the practice
problems for angles related to a circle
will come and join me in Chapter six ten
point six when we talked about more
angles as they relate to circles the
more angle our theorems in the next
edition of odd math