how are the 4 types of nitrogen-containing bases complementary to other bases?

The 4 bases are complementary to each other in the nature of their structure. For example, the natural structure of Cytosine and Guanine allow for three hydrogen bonds between each other without being inconsistently too short or long compared to the other pairings (this is why pyrimidines must be paired with purines, and not pyrimidines with pyrimidines or purines with purines). Just so you know, Adenine and Thymine have two hydrogen bonds with each other. This gives DNA that C-G and A-T base pair rule.

I'm pretty sure that the DNA structure is held in it's double helix form via hydrogen bonds. Hope this helped!