Question

in a rectangle ABCD BM and CN are perpendicular from B and D respectively on AC . Show that triangle BMC congurent to triangle DNA​

Answer 1

triangle BMC and triangle DNA

BC =AD(opposite sides of rectangle are equal)

Angle BCM=Angle MAD(Alternate angles)

Angle BMC =Angle DNA (BM and DN perpendicular to AC)

Therefore triangle BMC congruent to triangle DNA

Therefore BM =DN (cpct)

Answer 2

Given : ABCD is a rectangle  and  BM ⊥AC and  DN⊥AC

To prove: $\bigtriangleup BMC \cong \bigtriangleup DNA$

Solution :

we know that ABCD is a rectangle

and

BM ⊥AC and

DN⊥AC

then

$\angle BMC = 90^\circ \\\\\angle DNA = 90^\circ$

(angles of the rectangle )

in triangle BMC and DNA

BC = AD (opposite side of rectangle are equal )

$\angle DAN = \angle BCM$

$\angle ADN = 180^\circ - (\angle DAN)- (\angle DNA)\\\\\angle ADN = 180^\circ - (\angle BCM)- (\angle BMC)\\\\\angle ADN =\angle CBM$

thus ,

$\bigtriangleup BMC \cong \bigtriangleup DNA$

by (ASA property)

Hence proved

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