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The process of solving the seventh problem of mathematical square is very important. Thank you!
Extend CB, intercept BE=DN, and connect AE.

ABCD is a square,

∴AD=AB,∠D=∠ABC=∠ABE=90

AD = AB,BE=DN,∠D=∠ABE=90

∴△ABE≌△ADN(SAS)

∴S△ABE=S△ADN

AN=AE,∠DAN=∠BAE

∠∠ Dan+∠ Bam = ∠ Bud-∠ Mann =90 -45 =45

∴∠ BAE+∠ BAM = 45, that is ∠ EAM = ∠ Man = 45.

AE = AN,AM=AM,∠EAM=∠MAN

∴△AMN≌△AME

Then s △ AMN = s △ AME = s △ ABM+s △ Abe = s △ ABM+s △ ADN.