Can an isosceles triangle be a regular triangle?
Honestly, this depends upon how the person technically defines "isosceles" and "regular". For instance, if for "isosceles" they mean "two and only two sides the same length", then, of course, the answer is no. For, in a regular triangle, it has to have 3 sides of equal length.
But, if for "isosceles" they mean "two sides the same length", then, of course, the answer is yes. For, a regular has to have at least 2 sides the same length (in fact, having 3 sides the same length, so pick which 2 you want).
So, for this, whoever was your instructor for this, I would go back to their definition of isosceles and regular. If it says something about, for "isosceles", "only two sides of equal length", the answer is no. If it doesn't specify something like "only two sides of equal length", the answer is yes.
An isosceles triangle means the lengths of only two sides of the triangle are congruent. A regular triangle is where all of the sides of the triangle have to be congruent.
Therefore, an isosceles triangle can't be a regular triangle since all 3 sides have to be congruent for the triangle to be a regular triangle.
It depends on your definition of isoceles.
There are people who think that there are 3 types of triangles: Scalene, isoceles and equilateral (or regular.) By their definitions, a triangle can only fit into one category. This is because some people define an isoceles triangle as having exactly two equal sides. If this is true, an isoceles triangle can't also be regular, because the latter requires all three sides to be equal.
However, if you define an isoceles triangle as having at least two equal sides, then it can certainly also be regular because a triangle with all three sides being equal would fit both definitions.
You could call a regular triangle isosceles. However, this would be like calling a square a rectangle; while true, it isn't the best name for the shape.