Yes, it is true that an isosceles triangle can be acute or obtuse. As long as a triangle has two equal angles and two equal sides, it is isosceles. That situation can occur in both acute (all angles less than 900) and obtuse (one angle more than 900) triangles.
Examples are a triangle with the angles being 75-75-30 degrees (acute) and one having angles of 120-30-30 (obtuse). Of course the sides of the angles that are equal would also be equal, so both examples are of isosceles triangles.
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No, it is not true.
No
Not true. A triangle with angles of 90, 45 and 45 is a right angled isosceles triangle. If you take a square piece of paper and fold it in half - from corner to corner - you will get this shape.
An obtuse triangle must have two acute angles and these can be congruent.
Not at all.