One of the proofs to this problem is shown in the
Let's construct the additional squares indicated by dotted lines. It
is clear from the illustration that angle C is the sum of angles A and
D. Angle B equals angle D because they are corresponding angles of
similar right triangles (with the respective legs in the 1:2
proportion). That means B can be substituted for D, which
automatically makes the C equals the sum of A and B.