This question needs an auxiliary line, as shown in the figure.
Comment on BM, construct parallelogram BNDM, and connect AD.
BM = AC (known), and ND is translated from BM.
∴ND=AC
Ⅶ in △AMC and △DAN
ND=AC (confirmed)
Angle C= angle DNA
AN=MC (known)
∴△AMC≌△DAN (corner)
∴AN=AD (corresponding edge)
Angle 3= Angle 2 (corresponding angle)
In right triangle ACM, angle 1+ angle 2 = 180- angle C = 90.
∴ Angle 1+ Angle 3 = 90? Angle dam = 90.
∴ Angle AMD= Angle ADM=45 degrees.
BN‖MD again? ∴ Angle AMD= Angle BPM=45.
Finally solved ~ in fact, there are other ways.
Good luck ~