L14 : Prop of parallelogram - Quadrilaterals , Maths, Class 8 Video Lecture

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quadrilaterals part 14 is brought to you
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so now that we have discussed three
important types of quadrilateral let us
start discussing parallelogram in more
detail so we will talk about the
properties of parallelogram as I
mentioned before in parallelogram the
opposite sides are equal that is this is
equal to this I get let's name this that
makes it easier
so when you say opposite sides are equal
we mean a B is equal to C D and also ad
is equal to BC opposite angles are also
equal that is angle a is equal to angle
C and angle B is equal to angle D
adjacent angles are supplementary that
is sum of the adjacent angles is equal
to 180 degree now two angles are said to
be supplementary if their sum is equal
to 180 degrees so we can say that
looking at this picture angle A plus
angle D is equal to 180 degree or we can
say and then a plus angle D is equal to
180 degree or we can say angle B plus
angle C is equal to 180 so any two
adjacent angles their son will be equal
to 180 degrees diagonals bisect each
other so if you draw the diagonals of a
parallelogram so one diagonal divides it
into two triangles now if you draw both
the diagonals you see that the diagonals
bisect each other now hence you bisect
each other that means they meet at a
point such that this point divides the
diagonal into two equal halves so
basically this will be equal to this or
this will be equal to this so when we
say diagonals bisect each other that
would mean a o is equal to CE o and B
au is equal to do so that is the meaning
of bisection so and when a line is
bisecting another line that means it is
dividing that line into two equal halves
so these are the properties of a
parallelogram so now we will spend some
time trying to prove each of these
properties because I just told you that
these are the properties but why should
you trust me just like that so let us
try to mathematically prove each of
these properties so let us start with
opposite sides are equal how do we know
that opposite sides in a parallelogram
are equal so now in order to prove this
that the opposite sides in a
parallelogram are equal what we would do
is we would draw a diagonal to this
parallelogram and also let us name the
parallelogram let's say this is a b c d
a parallelogram where b d is a diagonal
so now what do we do now we will compare
these two triangles so let's see
triangle abd and triangle BCD so in
these two triangles what do we see now
in a parallelogram we know that the
opposite sides are parallel that is why
it is a parallelogram right so let's
also name these angles let us call this
angle 1 angle to angle 3 and angle 4 now
since these two sides since a B is
parallel to C D so that is something
which is known to us because that is a
part of the definition that
parallelogram is a quadrilateral with
opposite sides parallel the since they
are parallel therefore angle one will be
equal to angle four right the angle 1
and angle 4 will be equal similarly
angle 2 would be equal to angle 3
because BC is also parallel to
and when you compare these two triangles
you see that BD is equal to BD which is
a common side now considering these
three parameters what do you see that
two angles and one side are equal for
both these two triangles now as per our
knowledge on congruence of triangles we
can say that triangle abd is congruent
to triangle BCD by which rule this would
be a s a that is angle-side-angle
congruency where two angles and an aside
included by these angles are equal in
both the triangles so therefore these
two triangles are congruent now whenever
two triangles are congruent then all
their corresponding sides and all their
corresponding angles are also equal now
since these two triangles now this
triangle and this triangle are congruent
therefore this angle a will be equal to
this angle C therefore angle a would be
equal to angle C that is opposite angles
would be equal and also a B would be
equal to CB that is opposite sides will
be equal so this is what we wanted to
prove that opposite sides in a
parallelogram are equal and B proved it
let us now try to prove that opposite
angles are also equal in fact we almost
proved it in the previous slide but so
but okay let's see it once again so
again let's name the parallelogram it's
ABCDE we draw a diagonal easy
let's also name these angles let's say
this is angle 1 this is angle 2 this is
angle 3 and this is angle 4 now what are
the things that we know we know by
definition of parallelogram that a B is
parallel to CD and BC is parallel to
this is something which is known to us
and we want to prove that angle B is
equal to angle D they are the opposite
angles so we want to prove that angle B
is equal to angle D so what do we do
again we compare triangles a b c and
travel a d c and we find that since a b
and c d are parallel to each other
therefore and then 1 is equal to angle 2
since BC and AD are parallel to each
other therefore under 3 is equal to
angle 4 and AC is equal to AC which is a
common side so therefore considering
these three parameters we see that
triangle ABC is congruent to triangle
ABC by a s a congruence now since these
two triangles are congruent therefore
they are corresponding angles are also
equal therefore angle B is equal to
angle D now in case you have any
questions in your mind like this that ok
these two triangles are congruent by Y
but why did angle D become equal to
angle D that's because if you look at
the other angles angle 1 is already
equal to angle to angle 4 is already
equal to angle 3 so you are just left
with one angle in this triangle which is
angle B and you are left with one angle
in this triangle which is angle D so
therefore angle B will be equal to angle
D which are nothing but the opposite
angles of the parallelogram so this
proves that the opposite angles are
equal so let's now try to prove that the
adjacent angles are supplementary that
is sum of the adjacent angles is equal
to 180 degrees so for this again we
consider this parallelogram let us label
it as ABCD and here again we draw a
diagonal AC let's also label the angles
let's call this angle 1 this angle 2
this is angle 3 and the
angle four so now let us look at let us
consider any one triangle let us say we
consider the triangle ACD now we know
that sum of the three angles in a
triangle is equal to 180 degrees
therefore we can say angle 1 plus angle
4 plus angle T is equal to 180 degrees
right now when you look at this angle 1
is equal to angle 2 that's because this
side and this side a B is parallel to CD
therefore angle 1 will be equal to angle
2 alternate angles right so instead of
angle 1 we can very well write it as
angle 2 plus angle 4 plus angle D is
equal to 180 degree now what is angle 2
plus angle 4 that is nothing but angle
in there for angle A plus angle D is
equal to 180 degrees and what are these
angle a and angle D are nothing but
adjacent angles so this angle and this
angle these are adjacent angles the sum
of the adjacent angles is 180 degree
therefore they are supplementary in a
similar way you can also prove that
angle D plus angle C is also equal to
180 degrees and that's again very simple
like what we had written here angle 1
plus angle 4 plus angle D is equal to
180 degrees now instead of writing angle
1 is angle 2 we can also write angle for
this angle is equal to angle 3 so
instead of angle 4 we can write it as
angle 3 plus angle D is equal to 180 now
angle 1 plus angle 3 is nothing but
angle C so we proved that angle C plus
angle D is equal to 180 degrees
therefore the sum of any two adjacent
angles in a parallelogram is equal to
180 degrees let's move on to the fourth
property of the parallelogram that the
diagonals bisect each other so for this
again we have considered this
parallelogram ABCD and here again we
will draw the diagonal
this time we will draw both the
diagonals and let us mark this point
where the two diagonals meet as point O
so what are we trying to prove here in
in this case we want to prove that the
diagonals bisect each other that means
the diagonals intersect at a point such
that each diagonal gets divided into two
equal halves so we want to prove that a
o is equal to OC and B o is equal to Co
D so this is what you want to prove so
for that again let us consider these two
triangles angle a or triangle a OD and
triangle B or C because these are the
two triangles which involve these parts
a o OC d o and B ho and also let us mark
name these angles let us call this angle
one this is angle 2
this is angle 3 and this is angle 4 just
to make our derivation easier now in
these two triangles if you try to find
out something which are equal what do
you see the first thing that you see is
ad is equal to BC that's because
opposite sides in a parallelogram are
equal so that's something which we
already proved now opposite sides equal
next thing that we see is angle 2 is
equal to angle 3 this angle equal to
this angle because these are alternate
angles and we know that in a
parallelogram ad is equal to BC or
opposite sides are parallel in a similar
way angle 1 will be equal to angle 4
that is also because ad is equal to BC
it is parallel to BC now if you look at
these three criteria what do you see at
one side and two angles are equal for
both the triangles therefore you can say
that triangle a OD is congruent to
triangle B OC by a s a congruence now
once these two triangles are congruent
then what happens their corresponding
parts also become equal therefore what
happens d au is equal to b au and
ayoh is equal to OC and that is what we
wanted to prove so we can say hence
proved so therefore we very clearly
proved that the diagonals in a
parallelogram always bisect each other
so till now we have discussed about the
properties of parallelograms in fact the
four major properties of a parallelogram
and we have also proved them thank you
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