Triangles, Area, and Congruence

A couple of weeks ago we explored the areas of different pieces of a tangram. We also looked at congruent triangles in the tangram. A number of people said that one triangle was congruent to another triangle "because their areas are equal." Here is my question:

If two triangles have equal area, does that mean that they are congruent? If your answer is yes, explain why. If your answer is no, provide some counterexamples - that is, give me some triangles that have equal area but are not congruent.