For me this turned out to be sort of a challenging problem from the 2025 Math Calendar.
“Given equal line segments AB = CD, what is angle θ in degrees?”
See Elusive Angle for a solution
Meditations on Mathematics
A compendium of puzzles, problems, math inquiries, and math commentary.
Another method.
Let BC = 1. Hence CD=AB= sqrt(2)
Also angle CBD = 180-45=135 deg.
From triangle CBD,
sin(ang BDC) / BC = sin(ang CBD) / CD
sin(ang BDC) / 1 = sin(135) / sqrt(2)
sin(ang BDC) = 1/2
ang BDC= 30 deg
ang BCD = 180 – 135 – 30 = 15 deg.
********