Question

which of the following additional statements would allow us to prove that
\\( \triangle a b c \\) is equilateral?

choose 1 answer:

\\( \overline { a b } \cong \overline { b c } \\) only

\\( \overline { a b } \cong \overline { a c } \\) only

either statement is sufficient.

we need both statements.

even with both statements, we still could not prove that
\\( \triangle a b c \\) is equilateral.

Explanation:

Step1: Recall the definition of an equilateral triangle

An equilateral triangle has all three sides equal.

Step2: Analyze each option

• Option A: $\overline{AB}\cong\overline{BC}$ only. This makes $\triangle ABC$ isosceles with $AB = BC$, but we don't know about $AC$.
• Option B: $\overline{AB}\cong\overline{AC}$ only. This makes $\triangle ABC$ isosceles with $AB = AC$, but we don't know about $BC$.
• Option C: Either statement is not sufficient as explained above.
• Option D:If $\overline{AB}\cong\overline{BC}$ and $\overline{AB}\cong\overline{AC}$, then by transitivity $\overline{AB}\cong\overline{BC}\cong\overline{AC}$. So, $\triangle ABC$ is equilateral.
• Option E: This is incorrect as we can prove with both statements.

Answer:

D. We need both statements.