Question

determining if figures are congruent are △def and △rpq congruent? yes. △def can be mapped to △rpq by a reflection across the x - axis followed by a translation 2 units left. yes. △def can be mapped to △rpq by a 180° rotation about the origin followed by a translation 2 units down. no. △def cannot be mapped to △rpq by a rigid transformation or combination of rigid transformations.

Explanation:

Step1: Recall congruence by rigid - motion

Rigid motions (reflections, rotations, translations) preserve shape and size. We need to check if we can map $\triangle DEF$ to $\triangle RPQ$ using these.

Step2: Analyze the first option

A reflection across the $x$ - axis changes the sign of the $y$ - coordinates. A translation 2 units left subtracts 2 from the $x$ - coordinates. If we reflect $\triangle DEF$ across the $x$ - axis, the orientation will not match $\triangle RPQ$.

Step3: Analyze the second option

A $180^{\circ}$ rotation about the origin changes the signs of both $x$ and $y$ coordinates ($(x,y)\to(-x,-y)$). A translation 2 units down subtracts 2 from the $y$ - coordinates. If we rotate $\triangle DEF$ $180^{\circ}$ about the origin and then translate 2 units down, the orientation and position will not match $\triangle RPQ$.

Step4: Conclusion

Since we cannot map $\triangle DEF$ to $\triangle RPQ$ using a rigid transformation or a combination of rigid transformations, the triangles are not congruent.

Answer:

No. $\triangle DEF$ cannot be mapped to $\triangle RPQ$ by a rigid transformation or combination of rigid transformations.