Isosceles triangle
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No. I think it can. An isosceles triangle by definition has two equal sides. It is possible to draw a triangle with a right angle and two equal sides. The word isosceles is actually a Greek one standing for: having (two) equal sides. I thought that, in English, isosceles stands for "equilateral". If not, then yes it can be a triangle with two equal sides adjacent to the right angle. (hypotenuse excluded, anyway)
An isosceles triangle has two equal sides
A scalene triangle does not have any equal sides. Isosceles triangles have two equal sides
There is no specific name for a quadrangle with two equal sides.
Yes. Two adjacent sides are equal; the other two sides are also equal between themselves.