Algebra and Geometry both are derived from our understanding of numbers. Without one, the other would never have existed. The two mathematical studies evolved concurrently, because advances in Geometry would merit advances in Algebra and the same was true the opposite direction.
Geometry was the beginnings of Algebra. Through Geometry, mathematicians could visualize different problems that they had been working on. A good example is learning about different algebraic equations through the usage of squares and other geometric objects.
After many years, advances in Algebra started to lead the way to advances in Geometry. Through different equations and algebraic problems, different geometric figures could be constructed or at least found to at least exist.
Algebra and Geometry do not cause the discovery of the biggest ideas in either category today. Algebra also did not cause the discovery of the largest finding in Geometry. Through the continued practice of Geometric figures, non-Euclidean geometry was found by Gauss, Boylai, and Lobachevsky, all independently of one another. They found a contradictions to Playfair's Postulate.
While Algebra did not cause the biggest discovery in Geometry, I believe that Geometry did cause the discovery of the biggest Algebra discovery. By looking at triangles and other geometric figures (mostly squares), the discovery of Pythagorean's Theorem was made.