Question

In the given figure ABCD is a rectangle BM and DN are perpendicular to AC from B and D respectively. Prove that AN =CM ​

Answer 1

Step-by-step explanation:

Answer

ABCD is a parallelogram then,

AD∥BC and AC is a transversal.

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )In △BMC and △DNA

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )In △BMC and △DNA⇒ BC=AD [ Opposite sides of parallelogram are equal ]

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )In △BMC and △DNA⇒ BC=AD [ Opposite sides of parallelogram are equal ]⇒ ∠BCM=∠DAN [ From ( 1 ) ]

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )In △BMC and △DNA⇒ BC=AD [ Opposite sides of parallelogram are equal ]⇒ ∠BCM=∠DAN [ From ( 1 ) ]⇒ ∠BMC=∠DNA [ Both 90

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )In △BMC and △DNA⇒ BC=AD [ Opposite sides of parallelogram are equal ]⇒ ∠BCM=∠DAN [ From ( 1 ) ]⇒ ∠BMC=∠DNA [ Both 90 o

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )In △BMC and △DNA⇒ BC=AD [ Opposite sides of parallelogram are equal ]⇒ ∠BCM=∠DAN [ From ( 1 ) ]⇒ ∠BMC=∠DNA [ Both 90 o . ]

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )In △BMC and △DNA⇒ BC=AD [ Opposite sides of parallelogram are equal ]⇒ ∠BCM=∠DAN [ From ( 1 ) ]⇒ ∠BMC=∠DNA [ Both 90 o . ]∴ △BMC≅△DNA [ By SAA congruence rule ]

AD∥BC and AC is a transversal.∴ ∠BCA=∠DAC [ Alternate angles ]i.e. ∠BCM=∠DAN ---- ( 1 )In △BMC and △DNA⇒ BC=AD [ Opposite sides of parallelogram are equal ]⇒ ∠BCM=∠DAN [ From ( 1 ) ]⇒ ∠BMC=∠DNA [ Both 90 o . ]∴ △BMC≅△DNA [ By SAA congruence rule ]∴ BM=DN [ By CPCT ]

Answer 2

Answer:

In 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)