Examples: Angles made by a Transversal of Parallel Lines Video Lecture - Class 7

hello friends this video on lines and
angles part 15 is brought to you by exam
feel calm know about fear from exam so
based on whatever we have learned so far
we will quickly look at some of the
questions we used on intersecting lines
parallel lines and transversals question
number one stake the property that is
used in each of the following statements
if a parallel to B then angle one is
equal to angle five so let us see that
this is a this line is a this is B and
these two lines are parallel to each
other so this is how we denote parallel
lines that is the two lines saying that
a and B are parallel to each other then
angle one is equal to angle trial why
these two angles are equal because they
are corresponding angles so the first
property would be pair of corresponding
angles are equal so pair of
corresponding angles are equal
if angle 4 is equal to angle 6 so this
is angle 4 and this is and the 6 so
these are alternate interior angles so
since alternate interior angles are
equal therefore the lines are parallel
because alternate interior angles are
equal only when the lines are parallel
and then 4 plus angle 5 is equal to 180
degree so I will 4 plus angle Phi which
is this property this property is sum of
interior angles on the same side of
transversal is equal to 180 degrees on
same side of transversal is equal to 180
degrees so these are the three important
properties with regards to transversal
on parallel lines question number 2 in
the adjoining figure B is parallel to Q
find the unknown angles that means you
have to find
values of a b c d e and f so let us
start with the simplest one which can be
found most easily now as we know that
these two lines are parallel so this
angle is known so which would be the
corresponding angle for this angle so an
angle which lies over this line on this
side of the transversal above the line
so angle D is corresponding to angle 20
to 125 degree
therefore the value of D would be 125
degrees because they form a pair of
corresponding angles so we found out one
angle now when you look at angle B and
angle B these are vertically opposite
angles because you see this line this
line and this line they are intersecting
lines and B and D are opposite angles on
the intersecting lines therefore B will
also be equal to B that is equal to 125
degree because these are vertically
opposite angles now let us talk about a
now when you look at a and B they form
linear pair right because they are
located adjacent to each other they also
give a sum of 180 degrees therefore we
can say angle A plus angle B is equal to
180 degrees this is due to linear there
therefore a would be equal to 180 degree
minus B that is 180 degree minus 125
degree which is equal to 55 degree so we
found out a we found out B we found out
D so what would we see in a and C are
vertically opposite angles therefore C
will be equal to a will be equal to 55
degree that is because they form
vertically opposite angles so we now
know ABCD so now we are left with E and
F now E and F will always also be equal
to each other because
these are also vertically opposite
angles so these are the two intersecting
kinds one is this line and the other one
is this line so E and F will also be
equal but we have to find out at least
one value now when you look at angle E
and angle a can you establish in any
relation of course they are also
corresponding angles so angle E is equal
to angle a is equal to 55 degrees
because these are also corresponding
angles now I will e is 55 degrees and
then F will also equal to angle E that
is 55 degree because they form
vertically opposite angles so you see
just with these concepts of vertically
opposite angles corresponding angles we
could actually find out so many unknown
angles so we found that angle is 55
degree and then F is also 55 degree and
then E is also 55 degree because 125
degree C again is 55 degree and D again
is 125 degrees so that's how we could
find out all these patterns question
number three
in the given figure the arms of two
angles are parallel so each other arms
of two angles that means a B this
actually means that a B is parallel to
de and BC is parallel to EF if angle ABC
is equal to 70 degree then find angle D
G C so D G C would be this angle now
when you want to find angle D G see it
here you just focus on this part of the
image which shows that a B is parallel
to D E and DC acts as a transversal so
BC is a transversal on a B and D D that
is how it is so therefore you can say
that angle
DGC is equal to angle ABC why because
these are corresponding angles therefore
what is the value of angle ABC it is 70
degree therefore you can say that angle
DGC is also equal to 70 degree so the
first part is done so this angle is 70
degree now let us look at the second
part where we have to find out whether d
EF that is this angle now here you focus
on this part where BC is parallel to EF
that is given so here de acts as a
transversal so de is acting as a
transversal therefore which are the
corresponding angles here is this angle
and this angle therefore angle D G C is
equal to angle D EF because these are a
pair of corresponding angles now DG C we
have just now calculated as 70 degree
therefore anger d EF is also equal to 70
degrees question number 4 in the given
figures below decide whether n is
parallel to M so let us look at the
first figure so we have to find out if
these two are parallel now these two
would be parallel only when one of those
conditions are true either the
corresponding angles are equal or the
alternate angles are equal or the sum of
the angles on the same side of
transversal is 180 degree so one of
these has to be true so here we are
given 57 degree as this angle let us
call this angle is angle one so let us
try to find out angle one because if
these two are parallel then angle one
should be equal to 123 degrees because
these are corresponding angles so let us
try to find angle one now angle one plus
57 degree is equal to 180 degree that's
because they form linear pair therefore
angle one is equal to 180 degree minus
57 which is equal to 123 degree so this
angle one is 123 degree so this
the lentils and then these are
corresponding angles so we see that the
corresponding angles are equal so now
you might ask that why is it that only
if one of these property holds true the
lines are parallel that's bigger than
the corresponding angles are equal
the alternate angles will also be equal
and then the sum of the interior angles
on the same side of transversal would
also be 180 degree because you would
have seen like when we proved each of
these properties we made use of the
other property like when we wanted to
prove that the corresponding angles are
equal we made use of the property that
some of the interior angles on the same
side of transversal is one utility
similarly when we want you to prove that
the sum of the interior angles is 180
degree then we made use of the property
that alternate angles are equal so all
these properties are kind of interlinked
to each other so even if we are able to
prove one of these we can see that the
lines are parallel let us look at the
next figure so here we have to prove if
n is parallel to M or not by looking at
the figure itself we feel that they are
not parallel because you see the
distance between them is changing with
time so they will not be parallel but we
have to prove that mathematically so if
these two are parallel this would be the
transversal so for this 75 degree this
would be the corresponding angle so let
us call this angle 1 so angle 1 plus 75
degree that is 1 plus 75 degrees 180
degree because it forms linear pair
therefore angle 1 will be equal to 180
degree minus 75 degree which is 105 B so
basically this angle is 105 degree so
this angle and this angle are not equal
so therefore angle one is not equal to
75 degree therefore M is not parallel to
M now when the corresponding angles are
not equal the ordinary angles will also
not be equal and there
the lines would not be parallel so with
this we have reached towards the end of
this lesson on lines and angles and I
hope that this lesson would have helped
you so in geometry do all about a very
good observation and more practice
obviously so I would advise you to
practice more and more questions not
only from ncert textbooks you can pick
up other textbooks and start solving
questions and I am sure you gain a lot
of confidence so with this I will
conclude this lesson see you all on the
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