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To solve problem 3: In the isosceles triangle \(DEF\) with base \(EF = 8\) cm, and \(DK\) being the angle bisector of \(\angle EDF\), we need to find \(KF\), \(\angle EDF\), and \(\angle DKE\), given that \(\angle EDK = 36^\circ\). 1. **Determine \(\angle EDF\):** Since \(DK\) is the bisector of \(\angle EDF\), \(\angle EDK = \angle FDK = 36^\circ\). Therefore, \(\angle EDF = \angle EDK + \angle FDK = 36^\circ + 36^\circ = 72^\circ\). 2. **Calculate \(\angle DKE\):** The sum of angles in triangle \(DEF\) is \(180^\circ\). \[ \angle DEF + \angle DFE + \angle EDF = 180^\circ \] Since \(DEF\) is isosceles, \(\angle DEF = \angle DFE\). Let \(\angle DEF = \angle DFE = x\). Therefore, \[ x + x + 72^\circ = 180^\circ \] \[ 2x = 108^\circ \] \[ x = 54^\circ \] \(\angle DEF = \angle DFE = 54^\circ\). Since \(\angle DKE = \angle DFE\), \(\angle DKE = 54^\circ\). 3. **Using the Angle Bisector Theorem for \(KF\):** According to the angle bisector theorem, \( \frac{DK}{KF} = \frac{DE}{EF} \). Because we need \(KF\) and we have \(EF = 8\) cm, you might need additional information (like lengths or relationships) to calculate \(KF\) directly unless specified otherwise. Without additional measurements or context provided directly from the problem, finding \(KF\) purely based on given details isn't possible with certainty. However, the angle values align with common geometry procedures.