The ABC Toy Company is creating two similar pieces for a board game, as shown below. How could the toy maker confirm that the pieces are similar using the definition of similarity in terms of similarity transformations?

triangles GHI and JKL

Verify ∠G ≅ ∠J and ∠I ≅ ∠L by reflection.
Verify segment GH ≅ segment JK and segment GI ≅ segment JL by reflection.
Verify ∠G ≅ ∠J and ∠I ≅ ∠L by translation.
Verify segment GH ≅ segment JK and segment GI ≅ segment JL by translation.

Two triangles are similar if their corresponding angles are the same size. In similar triangles, corresponding sides are always in the same ratio.

From inspection of the given diagram (attached), ΔGHI ~ ΔJKL.

Therefore, corresponding angles are congruent:

∠G ≅ ∠J
∠H ≅ ∠K
∠I ≅ ∠L

Corresponding sides are in the same ratio:

GH : JK = HK : KL = GI = JL

A similarity transformation is:

a dilation, or
a rigid motion and a dilation.

As ΔJKL is clearly larger than ΔGHI, the corresponding sides (line segments) are not congruent.

ΔJKL and ΔGHI appear to be the same shape but a different size, so there is no need to reflect either triangle.

Therefore, to confirm that the pieces are similar using the definition of similarity in terms of similarity transformation, the toy maker should:

Verify ∠G ≅ ∠J and ∠I ≅ ∠L by translation.