Triangles ABE and BCD in the figure are isosceles. AB is the base for ABE, and BD is the base for BCD, with equal sides AE, EB, and BC, CD, respectively.
In the given figure, the information provided indicates that triangle BDE is a right-angled triangle, denoted by the right angle at vertex D. Additionally, it is mentioned that AE = EB = BA, implying that triangle ABE is an isosceles triangle with AB as its base and AE, EB as equal sides. Similarly, BC = DC suggests that triangle BCD is also an isosceles triangle with base BD and equal sides BC, CD.
Therefore, both triangle ABE and triangle BCD are isosceles triangles based on the given conditions. In an isosceles triangle, two sides are of equal length, and the angles opposite those sides are also equal. In this case, AE = EB implies angle ABE is equal, and BC = DC implies angle BCD is equal.
In summary, both triangle ABE and triangle BCD are isosceles triangles according to the given figure, where AB and BD are their respective bases, and AE, EB, CD are their equal sides.