Answer:
45°
Step-by-step explanation:
The triangles CPN and APM are isosceles, which gives rise to angle relationships that can be used to find the desired angle.
Let the angle at A be represented by α. Then the angle at C is the complementary angle (90° -α). The angles MPA and NPC are, respectively ...
∠MPA = (180° -α)/2 = 90° -(α/2) . . . . . . . base angle of ΔAPM
∠NPC = (180° -(90° -α))/2 = 45° +α/2 . . . base angle of ΔCPN
The sum of angles MPA, MPN, NPA is the linear angle APC, so we have ...
∠MPA +∠MPN +∠NPA = 180°
(90° -α/2) +∠MPN +(45° +α/2) = 180°
135° +∠MPN = 180°
∠MPN = 45°
