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 \(60^\circ\) and all sides equal. A scalene triangle has all sides of different lengths and all angles of different measures (since if sides are different, by the Law of Cosines, angles will be different). For two triangles to be similar, their corresponding angles must be equal (AA, SAS, SSS similarity criteria).

Step2: Analyze angle measures

In an equilateral triangle, all angles are \(60^\circ\). In a scalene triangle, all angles are distinct and not all \(60^\circ\) (because if all angles were \(60^\circ\), it would be equilateral, not scalene). So the angle measures of an equilateral triangle and a scalene triangle can never be equal.

Step3: Relate to similarity transformations

Similarity transformations (dilations and rigid transformations) preserve angle measures. Since the angle measures of an equilateral triangle and a scalene triangle can never match, we can never map one onto the other using only dilations and rigid transformations. Thus, an equilateral triangle is never similar to a scalene triangle, and we can never map one onto the other with those transformations.

Answer:

First blank: never
Second blank: never