Question

is (\triangle tuv) congruent to (\triangle tmn)? justify your answer using rigid motions.
yes. a rotation and a reflection can map (\triangle tuv) onto (\triangle tmn).
yes. a translation and a rotation can map (\triangle tuv) onto (\triangle tmn).
no. it is not possible to map (\triangle tuv) onto (\triangle tmn) using only rigid motions.

Explanation:

Step1: Recall the properties of rigid motions

Rigid motions (translation, rotation, reflection) preserve the shape and size of a figure. Two triangles are congruent if one can be mapped onto the other using rigid motions.

Step2: Analyze the given triangles

Looking at \(\triangle TUV\) and \(\triangle TMN\), we note that the corresponding sides and angles are equal (\(TU = TM\), \(TV = TN\), \(UV = MN\), \(\angle TUV=\angle TMN = 135^{\circ}\), \(\angle TUV=\angle TMN = 22^{\circ}\), \(\angle TUV=\angle TMN = 23^{\circ}\)).

Step3: Determine the rigid motions

A rotation about point \(T\) can align the sides \(TU\) and \(TM\), and then a reflection can map the remaining parts of \(\triangle TUV\) onto \(\triangle TMN\).

Answer:

Yes. A rotation and a reflection can map \(\triangle TUV\) onto \(\triangle TMN\).