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The following figure shows a triangle ABC in which AB=AC. M is a point on AB

and N is a point on AC such that BM=CN.

Prove that:

(i) AM=AN

(ii) 𝚫AMC ≅ 𝚫ANB

(iii) BN=CM

(iv) 𝚫BMC ≅ 𝚫CNB

Answer:

Solution

"hello students my name is pramit singh
and i welcome you all
to leader learning homework one of
india's best online classes
now today in this video we have a
question that is based on the topic of
congruent triangles
the question says the falling figure
shows a triangle abc in which a b
is equal to ac that means this side
is a b and this is ac they are equal
basically they are isosceles triangle
m is a point on a b
and n is a point on ac these are the two
points m and
n such that bm this bm
is equal to cn so prove that
first part am is equal to n so starting
from first
in triangle bm's amc that means a
m c triangle a
m c and triangle a
and b triangle a and b
we have a b is equal to ac that is given
a b is equal to ac that is given
similarly we have bn is equal to
cn that is also given
now if you check if we subtract equation
2
that means this is equation 2 and this
is equation 1.
if you subtract a b minus b n a b
minus a a b minus
b m this is b m b m is equal to c n
a b minus b m if you subtract a b minus
c m
b m what you will be getting you will be
getting am
and if you subtract a c minus c n what
you will be getting a n
so what we have to prove in the first
part that we have proved now
similarly com in the second part
what we have triangle amc
and triangle a and b a
m c
in triangle a m c and triangle
a n b
now which two sides are equal ac is
equal to a b that is given
similarly angle a is equal to angle a
that is common angle so this is side
angle a is equal to angle a common
and third part is am is equal to n that
we have proved
now if you check we have side angle side
so what what we have proved congruent
that is angle triangle
amc is congruent
to triangle a and b
so first part proof second part proof
now third part if you check
we have cm is equal to bn
cm is where this is cm cm is equal to bn
how
that is based on cpct
corresponding parts of congruent
triangles now in the last part we have
triangle bmc
and triangle c and b so in triangle
bmc and triangle
c and b we have bm is equal to cn that
is given
second part we have bc is equal to bc
common side this bc common base that
means
and the last part is cn is equal to bn
cm is equal to pn cn we have already
proved
cm is equal to bn
so again side triple s theorem we have
proved
triangle bmc
is congruent to triangle c and b
so we have proved all the four parts i
hope you have understood the concept
very clearly but still if you have any
doubt
do comment in the comment box and please
like and subscribe our channel for lido
thank you"

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