Question

triangle congruence: asa and aas
which rigid transformation would map △mzk to △qzk?
a reflection across the line containing zk
a reflection across the line containing mz
a rotation about point z
a rotation about point k

Explanation:

Step1: Analyze reflection across \( \overline{ZK} \)

A reflection across the line containing \( \overline{ZK} \) will map \( M \) to \( Q \) (since \( \angle MZK=\angle QZK = 90^{\circ}\) and \( ZK\) is common side). The two - triangle \( \triangle MZK\) and \( \triangle QZK\) are congruent by \( ASA\) (if we consider \( \angle MKZ=\angle QKZ\), \( ZK = ZK\), \( \angle MZK=\angle QZK\)) or \( AAS\) (if we consider the right - angles, a pair of equal non - included angles and the common side). A reflection is a rigid transformation.

Step2: Analyze reflection across \( \overline{MZ} \)

A reflection across the line containing \( \overline{MZ}\) will not map \( \triangle MZK\) to \( \triangle QZK\) because \( Q\) is not on the "mirror - image" side with respect to \( \overline{MZ}\).

Step3: Analyze rotation about point \( Z \)

A rotation about point \( Z\) : If we rotate \( \triangle MZK\) about \( Z\), unless the rotation angle is \( 180^{\circ}\) (but there is no indication from the congruence criteria \( ASA\) or \( AAS\) that a rotation is the transformation. Also, the side - angle - side relationships for congruence are better explained by reflection.

Step4: Analyze rotation about point \( K \)

A rotation about point \( K\) will not map \( \triangle MZK\) to \( \triangle QZK\) as it will not preserve the side - angle - side (or angle - side - angle) relationships in the way congruence (by \( ASA\) or \( AAS\)) is presented in the figure.

Answer:

a reflection across the line containing \( \overline{ZK} \)