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:

To prove \(\triangle ABC \sim \triangle NML\), we use triangle similarity criteria (AA, SAS, SSS, or transformation - based).

• For \(\angle B\cong\angle M\): We know \(\angle A=\angle N = 90^{\circ}\) (right angles). If \(\angle B\cong\angle M\), by AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle NML\) since two angles of one triangle are congruent to two angles of the other triangle.
• For "\(\triangle ABC\) is a right triangle": Just knowing \(\triangle ABC\) is a right triangle does not give us any information about the angles or sides of \(\triangle NML\) to establish similarity. So this is not sufficient.
• For "\(\triangle ABC\) was rotated and dilated by a scale factor between 0 and 1": Rotation and dilation (with a scale factor) are similarity transformations. A dilation (with any non - zero scale factor) and rotation will map \(\triangle ABC\) to a similar triangle. So if \(\triangle ABC\) is rotated and dilated to get \(\triangle NML\), then \(\triangle ABC\sim\triangle NML\).
• For "\(\triangle ABC\) was translated right and down": Translation is a rigid transformation (preserves shape and size, i.e., congruence). It does not change the size of the triangle, so it would prove congruence, not similarity (unless the scale factor of dilation is 1, but just translation alone is for congruence). So this is not for similarity.
• For \(\angle C\cong\angle L\): We know \(\angle A=\angle N=90^{\circ}\). If \(\angle C\cong\angle L\), by AA similarity criterion, \(\triangle ABC\sim\triangle NML\) (since two angles of one triangle are congruent to two angles of the other triangle).

Answer:

• \(\boldsymbol{\angle B\cong\angle M}\)
• \(\boldsymbol{\triangle ABC}\) was rotated and dilated by a scale factor between 0 and 1.
• \(\boldsymbol{\angle C\cong\angle L}\)