Question

recognizing necessary information for similarity
what additional information could be used to prove that $\triangle abc \sim \triangle nml$?
choose three correct answers.
$\angle c \cong \angle l$
$\angle b \cong \angle m$
$\triangle abc$ was translated right and down.
$\triangle abc$ is a right triangle.

Explanation:

Step1: Recall Triangle Similarity Criteria

To prove two triangles similar, we can use AA (Angle - Angle) criterion, SAS (Side - Angle - Side) similarity, or SSS (Side - Side - Side) similarity. For right - angled triangles (both \(\triangle ABC\) and \(\triangle NML\) are right - angled as \(\angle A = 90^{\circ}\) and \(\angle N=90^{\circ}\)), if we can show one more pair of corresponding angles are equal, we can use AA similarity.

Step2: Analyze Each Option

• Option 1: \(\angle C\cong\angle L\)

Since \(\angle A=\angle N = 90^{\circ}\) (right angles) and \(\angle C\cong\angle L\), by AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle NML\). So this option is correct.

• Option 2: \(\angle B\cong\angle M\)

We know that \(\angle A=\angle N = 90^{\circ}\). If \(\angle B\cong\angle M\), then the third angles \(\angle C\) and \(\angle L\) will also be equal (since the sum of angles in a triangle is \(180^{\circ}\)). By AA similarity criterion, \(\triangle ABC\sim\triangle NML\). So this option is correct.

• Option 3: \(\triangle ABC\) was translated right and down.

A translation is a rigid transformation. Rigid transformations (translation, rotation, reflection) preserve the shape and size of the figure, so the image after translation is congruent (and thus similar, with a scale factor of 1) to the original figure. So if \(\triangle ABC\) is translated to get \(\triangle NML\), then \(\triangle ABC\sim\triangle NML\). So this option is correct.

• Option 4: \(\triangle ABC\) is a right triangle.

We already know that \(\triangle ABC\) is a right triangle (from the diagram, \(\angle A = 90^{\circ}\)), and we also know that \(\triangle NML\) is a right triangle ( \(\angle N=90^{\circ}\)). Just knowing that both are right triangles is not enough to prove similarity, we need more information about the angles or sides. So this option is incorrect.

Answer:

The three correct answers are:

• \(\boldsymbol{\angle C\cong\angle L}\)
• \(\boldsymbol{\angle B\cong\angle M}\)
• \(\boldsymbol{\triangle ABC}\) was translated right and down.