Final answer:
To prove triangle ABC is isosceles, you need information indicating that either two sides are equal or two angles are equal. Options b (two sides are equal) and c (two angles are equal) are both sufficient proofs for this. Option a refers to an equilateral triangle, which is a specific type of isosceles triangle, and option d is a general property of all triangles.
Step-by-step explanation:
To prove that triangle ABC is isosceles, you need information about its sides or angles. An isosceles triangle is defined as a triangle that has at least two sides of equal length, which subsequently also means that at least two of its angles are equal because of the base angles theorem. Therefore, here are the considerations for each option:
- a. All three angles are equal: This would mean the triangle is equilateral rather than isosceles; however, an equilateral triangle is a special case of an isosceles triangle since it has not just two, but all three equal sides.
- b. Two sides are equal: This directly meets the definition of an isosceles triangle and therefore is sufficient proof.
- c. Two angles are equal: This too is sufficient proof as it implies, by the base angles theorem, that the sides opposite those angles must be equal.
- d. The sum of interior angles is 180°: This fact is true for all triangles and does not provide specific information to prove that a triangle is isosceles.
As a result, options b and c are sufficient to prove that triangle ABC is an isosceles triangle. If you have either two equal sides or two equal angles, the triangle can be determined to be isosceles.