Question

complete the similarity statement.
an equilateral triangle is sometimes/always/never similar to a scalene triangle, because we can sometimes/always/never map one onto the other using only dilations and rigid transformations.

Explanation:

Step1: Recall triangle similarity and types

An equilateral triangle has all angles equal to \(60^\circ\) (all sides equal). A scalene triangle has all sides (and thus angles) of different measures (no equal angles). For triangles to be similar, their corresponding angles must be equal (AA similarity criterion, or via dilations/rigid transformations which preserve angle measures and scale side lengths).

Step2: Analyze angle measures

Since an equilateral triangle has three \(60^\circ\) angles, and a scalene triangle has three distinct angles (none equal to each other, and not all \(60^\circ\) because if all were \(60^\circ\), it would be equilateral, not scalene), their angle measures can never match. So we can never map one to the other with dilations (which scale, preserving angle) and rigid transformations (which preserve angle and length).

Answer:

First blank: never
Second blank: never