Question

recognizing necessary information for similarity
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

Two triangles are similar if their corresponding angles are equal (AA - Angle - Angle criterion). Also, a rotation (which is a rigid transformation that preserves angles) followed by a dilation (which changes the size but not the shape, i.e., preserves angles) can map one similar triangle to another.

Step2: Analyze each option

• For $\angle B=\angle M$:

If we know that $\angle A=\angle N = 90^{\circ}$ (from the right - angle markings) and $\angle B=\angle M$, then by the AA (Angle - Angle) similarity criterion, $\triangle ABC\sim\triangle NML$.

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

We already know from the figure that $\triangle ABC$ is a right triangle (since $\angle A = 90^{\circ}$) and $\triangle NML$ is a right triangle ($\angle N=90^{\circ}$). But just knowing they are right - triangles is not sufficient for similarity. There are infinitely many non - similar right - triangles.

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

A rotation is a transformation that preserves angle measures. A dilation (with a scale factor $k
eq0$) is a transformation that preserves the shape (i.e., angle measures). If $\triangle ABC$ is rotated (so that the corresponding angles are in the same relative position) and dilated (which does not change the angle measures), then $\triangle ABC\sim\triangle NML$.

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

A translation is a rigid transformation. Translating $\triangle ABC$ right and down will make it congruent (if no dilation) or still just a translated figure. Congruent triangles are a special case of similar triangles (with scale factor $k = 1$), but the problem is about proving similarity in general. Also, we don't know if there is a dilation involved with the translation. Just translation (a rigid motion) is not enough to prove similarity in the non - congruent sense.

• For $\angle C=\angle L$:

If we know that $\angle A=\angle N = 90^{\circ}$ (from the right - angle markings) and $\angle C=\angle L$, then by the AA (Angle - Angle) similarity criterion, $\triangle ABC\sim\triangle NML$.

Answer:

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