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 Similar Triangle Criteria

Similar triangles can be proven by AA (Angle - Angle), SAS (Side - Angle - Side), SSS (Side - Side - Side) criteria. Also, similarity transformations (rotation, dilation, translation, reflection) preserve similarity.

Step2: Analyze Each Option

• Option 1: $\angle B\cong\angle M$

We know $\angle A$ and $\angle N$ are right angles ($\angle A=\angle N = 90^{\circ}$). If $\angle B\cong\angle M$, then by AA (since two angles are congruent), $\triangle ABC\sim\triangle NML$.

• Option 2: $\triangle ABC$ is a right triangle.

Just knowing $\triangle ABC$ is a right triangle doesn't tell us about $\triangle NML$ or the angles/sides relationship between the two triangles. So this can't prove similarity.

• Option 3: $\triangle ABC$ was rotated and dilated by a scale factor between 0 and 1.

Rotation and dilation are similarity transformations. Dilation (with any non - zero scale factor) and rotation preserve the shape (angles remain the same, sides are scaled proportionally). So if a triangle is rotated and dilated to get another triangle, they are similar.

• Option 4: $\triangle ABC$ was translated right and down.

Translation is a rigid transformation (preserves shape and size), but it just moves the triangle. However, translation alone doesn't imply similarity in the sense of proving $\triangle ABC\sim\triangle NML$ (since we need to relate the two triangles' angles/sides. But actually, translation preserves congruence, and congruent triangles are similar. Wait, but let's check the triangles. Wait, the orientation: $\triangle ABC$ has right angle at $A$, $\triangle NML$ has right angle at $N$. If we translate $\triangle ABC$ right and down, the angles remain the same. But maybe the key is that rotation + dilation is a similarity transformation, translation is rigid. But let's re - evaluate. Wait, the triangles have different orientations. So translation alone may not map $\triangle ABC$ to $\triangle NML$ because of the right angle position. So this option is not correct for proving similarity (since the triangles are not just translated, there is a rotation involved in their orientation).

• Option 5: $\angle C\cong\angle L$

We know $\angle A=\angle N = 90^{\circ}$. If $\angle C\cong\angle L$, then by AA (two angles congruent), $\triangle ABC\sim\triangle NML$.

Answer:

$\angle B\cong\angle M$, $\triangle ABC$ was rotated and dilated by a scale factor between 0 and 1, $\angle C\cong\angle L$ (i.e., the options: $\boldsymbol{\angle B \cong \angle M}$, $\boldsymbol{\triangle ABC \text{ was rotated and dilated by a scale factor between 0 and 1}}$, $\boldsymbol{\angle C \cong \angle L}$)