1. **Problem statement:** We have a regular pentagon ABCDE. - AP is the angle bisector of \(\angle BAE\). - CQ is the angle bisector of \(\angle BCD\). We need to find the ratio \(x : y : z\) where \(x = AP\), \(y = PQ\), and \(z = QE\). 2. **Properties of a regular pentagon:** - All sides are equal. - All interior angles are \(108^\circ\). 3. **Analyze angle bisectors:** - Since AP bisects \(\angle BAE\), it divides the \(108^\circ\) angle at A into two \(54^\circ\) angles. - Similarly, CQ bisects \(\angle BCD\), dividing the \(108^\circ\) angle at C into two \(54^\circ\) angles. 4. **Using symmetry and equal sides:** - Because ABCDE is regular, segments AE and CD are equal. - Points P and Q lie on AE and CD respectively, determined by the angle bisectors. 5. **Express lengths in terms of side length \(s\):** - Let the side length of the pentagon be \(s\). - Using the Law of Sines and properties of angle bisectors in triangles formed, we find: \[ \frac{x}{z} = \frac{AP}{QE} = 1 \] - This is because AP and QE are symmetric angle bisectors in equal sides. 6. **Find \(y\) in terms of \(x\) and \(z\):** - Since \(PQ\) connects points on AE and CD, and given the symmetry, \(y = PQ = x\). 7. **Final ratio:** \[ x : y : z = 1 : 1 : 1 \] **Answer:** \(x : y : z = 1 : 1 : 1\)