1. **Problem Statement:** Given two parallelograms ABCD and ECBF, with $\angle A = 54^\circ$ in ABCD and $\angle B = 66^\circ$ in ECBF, find the value of $\angle ECF$. 2. **Recall properties of parallelograms:** - Opposite angles are equal. - Adjacent angles are supplementary, meaning their sum is $180^\circ$. 3. **Analyze parallelogram ABCD:** - Since $\angle A = 54^\circ$, the opposite angle $\angle C = 54^\circ$. - Adjacent angles $\angle A + \angle B = 180^\circ$, so $\angle B = 180^\circ - 54^\circ = 126^\circ$. - Similarly, $\angle D = 126^\circ$. 4. **Analyze parallelogram ECBF:** - Given $\angle B = 66^\circ$ in ECBF, the opposite angle $\angle F = 66^\circ$. - Adjacent angles sum to $180^\circ$, so $\angle C = 180^\circ - 66^\circ = 114^\circ$. 5. **Find $\angle ECF$:** - Point C is common to both parallelograms. - From ABCD, $\angle C = 54^\circ$, from ECBF, $\angle C = 114^\circ$. - The angle $\angle ECF$ is the angle at C between points E and F in parallelogram ECBF, which is $114^\circ$. **Final answer:** $$\boxed{66^\circ}$$ **Explanation:** The angle $\angle ECF$ corresponds to the angle adjacent to $\angle B = 66^\circ$ in parallelogram ECBF, which is $66^\circ$.