Question

what additional information could be used to prove that
△abc ~ △nml? check all that apply
□ ∠b ≅ ∠m
□ △abc is a right triangle
□ △abc was rotated and dilated by a scale factor
between 0 and 1
□ △abc was translated right and down
□ ∠c ≅ ∠l

Explanation:

Step1: Recall similarity criteria

For two right - angled triangles ($\angle A=\angle N = 90^{\circ}$), if we have another pair of congruent angles (by AA similarity criterion) or information about similarity - transforming (rotation and dilation which are similarity transformations) the triangles.

Step2: Analyze each option

• For $\angle B\cong\angle M$:

Since $\angle A=\angle N = 90^{\circ}$, by the AA (angle - angle) similarity criterion ($\angle A=\angle N$ and $\angle B\cong\angle M$), $\triangle ABC\sim\triangle NML$.

• For $\triangle ABC$ is a right triangle:

We already know from the figure that $\angle A = 90^{\circ}$ (so $\triangle ABC$ is a right - triangle) and $\angle N=90^{\circ}$ (so $\triangle NML$ is a right - triangle). But just knowing they are right - triangles is not enough for similarity (we need more angle - angle or side - angle - side in a proportion etc. information).

• For $\triangle ABC$ was rotated and dilated by a scale factor between $0$ and $1$:

Rotation is a rigid transformation (preserves shape) and dilation (with a non - zero scale factor) is a similarity transformation. If $\triangle ABC$ was rotated and dilated (a composition of similarity transformations) to get $\triangle NML$, then $\triangle ABC\sim\triangle NML$.

• For $\triangle ABC$ was translated right and down:

Translation is a rigid transformation (preserves size and shape). If only translation is done, the triangles will be congruent (not just similar, and we have no information about side - length ratios).

• For $\angle C\cong\angle L$:

Since $\angle A=\angle N = 90^{\circ}$, by the AA (angle - angle) similarity criterion ($\angle A=\angle N$ and $\angle C\cong\angle L$), $\triangle ABC\sim\triangle NML$.

Answer:

• $\angle B\cong\angle M$
• $\triangle ABC$ was rotated and dilated by a scale factor between $0$ and $1$
• $\angle C\cong\angle L$