Triangles having the same base have equal area.

Concept: To calculate the area of a parallelogram, multiply the perpendicular's base by its height.

It should be noted that the parallelogram's base and height are perpendicular to each other, but the parallelogram's lateral side is not. 

It is false.

$Area of triangle =\frac{1}{2}\times base\times height$

From the formula, it is clear that the area of triangles depends on base and height and not only on the base.

So, triangles having same base may not have same area because of different height