Question

Which of the following additional statements would allow us to prove that △ ABC is equilateral?

a. AB≌BC only
b. AB≌AC only
c. Either statement is sufficient.
d. We need both statements.
e. Even with both statements, we still could not prove that △ ABC is equilateral.

Answer 1

To prove ∆ ABC is equilateral, we need both statements a and b, demonstrating that AB≡BC and AB≡AC, to show all sides are congruent by transitive property. Thus, the correct answer is d. We need both statements.

Step-by-step explanation:

To prove that a triangle is equilateral, we need to show that all three sides are congruent to each other. An equilateral triangle not only has three congruent sides, but also three congruent angles, each measuring 60 degrees, which is a characteristic property of equilateral triangles.

In the case of ∆ ABC, if we are given that AB≡BC (statement a) or AB≡AC (statement b), we are only demonstrating the congruence of two sides. These pieces of information are individually insufficient to conclude that ∆ ABC is equilateral, as it is possible to construct isosceles triangles where only two sides are congruent.

However, if we are provided with both statements (meaning AB≡BC and AB≡AC), then by transitive property (if AB≡BC and AB≡AC, then BC≡AC), all three sides are shown to be congruent, allowing us to determine that the triangle is indeed equilateral. Thus, the correct answer is d. We need both statements.