Question

1. triangle abd is a rotation of triangle cdb around point e by 180°. is angle abd congruent to angle cdb? if so, explain your reasoning. if not, which angle is adb congruent to?
2. polygon hijkl is a reflection and translation of polygon gfonm. is segment kj congruent to segment nm? if so, explain your reasoning. if not, which segment is nm congruent to?
3. quadrilateral pqrs is a rotation of polygon vzyw (not pictured). is angle qrs congruent to angle zyw? if so, explain your reasoning. if not, which angle is qrs congruent to?

Explanation:

Step1: Recall congruence after rotation

A 180 - degree rotation is a rigid transformation. Rigid transformations preserve angle - measure and side - length. For $\triangle ABD$ and $\triangle CDB$ which are related by a 180 - degree rotation around point $E$, corresponding angles are congruent. $\angle ADB$ corresponds to $\angle CBD$. So $\angle ABD$ is not congruent to $\angle CDB$.

Step2: Recall congruence after reflection and translation

Reflection and translation are rigid transformations. For polygon $HIJKL$ which is a reflection and translation of polygon $GFONM$, corresponding sides are congruent. Segment $NM$ corresponds to segment $KL$, so segment $KJ$ is not congruent to segment $NM$.

Step3: Recall congruence after rotation

Since quadrilateral $PQRS$ is a rotation of polygon $VZYW$, corresponding angles are congruent. $\angle QRS$ corresponds to $\angle ZYW$.

Answer:

1. $\angle ADB$ is congruent to $\angle CBD$.
2. Segment $NM$ is congruent to segment $KL$.
3. $\angle QRS$ is congruent to $\angle ZYW$.