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Byju's Answer
Standard IX
Mathematics
ASA Criteria for Congruency
In the given ...
Question
In the given figure, ABCD is a square and EF is parallel to diagonal DB and EM = FM. Prove that: (i) BF = DE (ii) AM bisects
∠BAD.
Figure
Open in App
Solution
i
We
have
,
C
D
=
B
C
-
-
-
-
-
-
-
1
S
ides
of
a
square
are
equal
and
∠
B
D
C
=
∠
D
B
C
=
45
°
Given
:
E
F
∥
D
B
∴
∠
B
D
C
=
∠
F
E
C
C
orresponding
angles
∴
∠
F
E
C
=
45
°
Similarly
,
∠
E
F
C
=
45
°
∴
△
C
E
F
is
an
isosceles
triangle
.
and
C
E
=
C
F
-
-
-
-
-
-
-
2
From
1
-
2
,
we
get
:
C
D
-
C
E
=
B
C
-
C
F
⇒
D
E
=
B
F
∴
B
F
=
D
E
ii
In
△
A
B
F
and
△
A
D
E
,
A
B
=
A
D
All
sides
of
a
square
are
equal
∠
A
B
F
=
∠
A
D
E
Both
angles
are
equal
to
90
°
B
F
=
D
E
(
Proved
above
)
∴
△
A
B
F
≅
△
A
D
E
by
S
A
S
congruency
criteria
∴
∠
B
A
F
=
∠
D
A
E
-
-
-
-
3
And
A
F
=
A
E
-
-
-
-
-
-
4
Now
,
In
△
A
M
E
and
△
A
M
F
,
A
M
=
A
M
C
ommon
side
E
M
=
F
M
G
iven
A
E
=
A
F
from
4
∴
△
A
M
E
≅
△
A
M
F
By
SSS
congruency
criteria
∴
∠
E
A
M
=
∠
F
A
M
∴
∠
F
A
M
=
∠
E
A
M
-
-
-
-
-
5
From
3
+
5
,
we
get
:
∠
B
A
F
+
∠
F
A
M
=
∠
D
A
E
+
∠
E
A
M
∠
B
A
M
=
∠
D
A
M
Therefore
,
A
M
bisects
∠
B
A
D
.
Suggest Corrections
10
Similar questions
Q.
In the figure given alongside, ABCD is a square and EF is parallel to diagonal BD. If EM = FM, prove that DF = BE.
Q.
In the given figure, BA
⊥ AC and DE ⊥ EF such that BA =DE and BF = DC. Prove that AC = EF.
