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:

Analyze properties of equilateral and scalene triangles

An equilateral triangle has three equal sides and three equal angles (all \(60^\circ\)). A scalene triangle has three sides of different lengths and three different angles. Since similar triangles must have congruent corresponding angles, an equilateral triangle (with angles \(60^\circ, 60^\circ, 60^\circ\)) can never have congruent angles to a scalene triangle (which cannot have three equal angles). Thus, an equilateral triangle is never similar to a scalene triangle.

Analyze transformation mapping

Similar figures can be mapped onto one another using a sequence of rigid transformations and dilations, which preserve angle measures and shape. Since an equilateral triangle and a scalene triangle are never similar, we can never map one onto the other using only dilations and rigid transformations.

Answer:

An equilateral triangle is <blank>never</blank> similar to a scalene triangle, because we can <blank>never</blank> map one onto the other using only dilations and rigid transformations.