Question: 4 In the diagram, ABDE is a trapezium and CDEF is a parallelogram. Given that BC = CF and $\angle CDE = 123^\circ$, stating the reasons clearly, find (a) angle CBF, Answer $\angle CBF = ...............^\circ$ [2] (b) angle BFE. Answer $\angle BFE = ...............^\circ$ [1]

1. **Problem Statement:** We are given a trapezium ABDE and a parallelogram CDEF with $BC = CF$ and $\angle CDE = 123^\circ$. We need to find $\angle CBF$ and $\angle BFE$. 2. **Key Properties and Formulas:** - In a parallelogram, opposite angles are equal. - Adjacent angles in a parallelogram are supplementary (sum to $180^\circ$). - Since $BC = CF$, triangle BCF is isosceles. - The sum of angles in a triangle is $180^\circ$. 3. **Find $\angle CBF$:** - Given $\angle CDE = 123^\circ$ and CDEF is a parallelogram, $\angle CDE = \angle CFE = 123^\circ$ (opposite angles in parallelogram). - Since $\angle CFE = 123^\circ$, adjacent angle $\angle BFE = 180^\circ - 123^\circ = 57^\circ$ (linear pair). - Triangle BCF is isosceles with $BC = CF$, so $\angle CBF = \angle CFB$. - Sum of angles in triangle BCF: $$\angle CBF + \angle CFB + \angle BCF = 180^\circ$$ $$2 \times \angle CBF + 57^\circ = 180^\circ$$ $$2 \times \angle CBF = 180^\circ - 57^\circ = 123^\circ$$ $$\angle CBF = \frac{123^\circ}{2} = 61.5^\circ$$ 4. **Find $\angle BFE$:** - From step 3, $\angle BFE = 57^\circ$. **Final answers:** - $\angle CBF = 61.5^\circ$ - $\angle BFE = 57^\circ$