Question

1. how would you prove that the triangle below is equilateral?

show all angles to be congruent.
show at least two sides to be congruent.
show all sides to be congruent.
show two sides are of equal measure.

Explanation:

An equilateral triangle is defined as a triangle with all three sides of equal length (congruent). If all sides are congruent, then by the SSS (Side - Side - Side) congruence criterion and the properties of triangles, all angles will also be congruent (each equal to \(60^{\circ}\)).
Showing at least two sides congruent (or two sides of equal measure) would make it an isosceles triangle, not necessarily equilateral. While showing all angles congruent (by the converse of the base angles theorem and angle - sum property of triangles) can also imply an equilateral triangle, the most direct and fundamental definition is based on side lengths.

Answer:

Show all sides to be congruent.