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 sides equal (all angles \(60^\circ\)). A scalene triangle has all sides of different lengths (and all angles different). For two triangles to be similar, their corresponding angles must be equal (AA, SAS, SSS similarity criteria). Since an equilateral triangle's angles are all \(60^\circ\), and a scalene triangle's angles are all different (and not all \(60^\circ\) unless it's equilateral, but scalene can't be equilateral), their angle measures can't match. So, we can never map one onto the other via dilations (which preserve angle measures) and rigid transformations (which also preserve angle measures) to make them similar.

Step2: Determine the first blank

Since the angle measures of equilateral and scalene triangles can never be equal (equilateral has all \(60^\circ\), scalene has all different angles, not all \(60^\circ\)), an equilateral triangle is never similar to a scalene triangle.

Step3: Determine the second blank

Because the angle measures don't match, we can never map one onto the other using dilations (which keep angles the same) and rigid transformations (which also keep angles the same) to achieve similarity. So the second blank is "never".

Answer:

First blank: never
Second blank: never