Question

All isosceles triangles are similar. Two angles within each triangle are always congruent.
All isosceles triangles are similar. The triangle sum theorem states that the sum of the angles in a triangle is 180°. Therefore, the third angle can always be determined.
All isosceles triangles are not similar. The pair of congruent angles within one triangle is not necessarily congruent to the pair of congruent angles within the other triangle.
All isosceles triangles are not similar. Given only the vertex angle of an isosceles triangle, there is not enough information to determine the measures of the base angles. Therefore, it is not possible to determine if the base angles of one isosceles triangle are congruent to the base angles of another.

Answer 1

Short answer: Not true in general. All isosceles triangles are not necessarily similar.

Why: Similarity of triangles requires all corresponding angles to be equal. An isosceles triangle only guarantees two equal angles (the base angles), but their measures depend on the vertex angle. If the vertex angle is θ, the base angles are (180° − θ)/2. Different isosceles triangles can have different θ, so their angles (and thus their shapes) can differ.

Example:
- Isosceles A: vertex = 40°, base angles = 70°, 70°.
- Isosceles B: vertex = 20°, base angles = 80°, 80°.
A and B are not similar.

Correct refined statement: Any two isosceles triangles with the same vertex angle (equivalently, the same base-angle measure) are similar. But simply knowing "isosceles" without the actual angle does not guarantee similarity.