Make the symmetry points e and f of point a about BC and CD respectively

Connect EF to BC and CD at points m and n.

AM = EM，AN=FN

∴△AMN perimeter =AM+MN+AN=EM+MN+FN=EF

∴EF is the minimum value of the circumference of △AMN.

At this time: AB=EB, AD=FD.

According to the triangle exterior angle theorem are:

∠AMN=2∠DAN

∠ANM = 2∠ bam

Add the above two formulas:

∠AMN+∠ANM

=2(∠DAN+∠BAM)

= 2(∠ bad-∠ people)

=2∠BAD-2( 180 -∠AMN+∠ANM)

=2× 120 -360 +2(∠AMN+∠ANM)

=2(∠AMN+∠ANM)- 120

Solution: ∠ AMN+∠ ANM = 120.