Question

is △abc congruent to △def? justify your answer using rigid motions.
yes. a reflection can map △abc onto △def.
yes. a translation can map △abc onto △def.
no. it is not possible to map △abc onto △def using only rigid motions.

Explanation:

Step1: Recall the properties of rigid motions

Rigid motions (reflection, translation, rotation) preserve the shape and size of a figure. For two triangles to be congruent via rigid motions, their corresponding sides and angles must be equal.

Step2: Analyze the given triangles

In \(\triangle ABC\) and \(\triangle DEF\), we check the correspondence of sides and included angles. The side - angle - side (SAS) congruence criterion: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
Here, \(AB = DE=12m\), \(AC = DF = 5m\), and \(\angle A=\angle D\) (the included angles). A reflection (a type of rigid motion) can map \(\triangle ABC\) onto \(\triangle DEF\) because reflection preserves the lengths of sides and measures of angles. A translation alone cannot map \(\triangle ABC\) onto \(\triangle DEF\) as the orientation (due to the position of the vertices relative to each other) is better achieved by a reflection.

Answer:

Yes. A reflection can map \(\triangle ABC\) onto \(\triangle DEF\).