1. **Problem statement:** Given that lines AB // CD and BP // AD, with angles \(\angle EBF = 68^\circ\) and \(\angle CDE = 58^\circ\), find (i) \(\angle AEB\) and (ii) \(\angle ABE\). 2. **Relevant properties and formulas:** - When two lines are parallel, alternate interior angles are equal. - The sum of angles around a point is \(360^\circ\). - The sum of angles in a triangle is \(180^\circ\). 3. **Step (i) Find \(\angle AEB\):** - Since AB // CD and BP // AD, quadrilateral ABPD is a parallelogram. - \(\angle EBF = 68^\circ\) is given. - \(\angle CDE = 58^\circ\) is given. - Using the parallel lines and transversal properties, \(\angle AEB\) corresponds to \(\angle CDE\) because of alternate interior angles. - Therefore, \(\angle AEB = 58^\circ\). 4. **Step (ii) Find \(\angle ABE\):** - In triangle ABE, angles are \(\angle ABE\), \(\angle AEB = 58^\circ\), and \(\angle EBF = 68^\circ\). - Since \(\angle EBF = 68^\circ\) is adjacent to \(\angle ABE\), and BP // AD, \(\angle ABE = \angle EBF = 68^\circ\) (corresponding angles). 5. **Final answers:** - \(\angle AEB = 58^\circ\) - \(\angle ABE = 68^\circ\)