Final answer:
The music popular in the second half of the 1800s that featured a(repeated)b(repeated)cdc is known as the Rondo form.
Step-by-step explanation:
The type of music popular in the second half of the 1800s that featured a(repeated)b(repeated)cdc is known as the Rondo form. Rondo is a musical form characterized by the repeated return of a main theme, interspersed with contrasting sections. In this case, the main theme is represented by the pattern a(repeated)b(repeated)cdc.
One example of a composition that follows this form is Ludwig van Beethoven's Piano Sonata No. 8 in C minor, Op. 13, commonly known as the 'Pathétique' Sonata. The second movement of this sonata features the a(repeated)b(repeated)cdc pattern.