Final answer:
A triangle ABC with two sides of length 'a' can be either an isosceles triangle or an equilateral triangle, depending on the length of the third side and if it obeys the triangle inequality theorem.
Step-by-step explanation:
If a triangle ABC has two sides of length 'a', it can be either an isosceles triangle or an equilateral triangle. To determine which types of triangles are possible, we must think about the properties of triangles. Firstly, every triangle has three sides and the sum of its angles is always 180 degrees.
In an isosceles triangle, at least two sides are of equal length. Therefore, if two sides are of length 'a', the third side could be any length as long as it adheres to the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If the two sides of length 'a' are adjacent, the triangle is isosceles with two sides and two angles equal.
If all three sides of triangle ABC are of equal length 'a', then ABC is an equilateral triangle, which is a special case of an isosceles triangle where all three sides and angles are equal, and each angle measures 60 degrees.