1. **Problem statement:** Find the locus of points inside quadrilateral ABCD that are equidistant from sides AB and BC. 2. **Key concept:** The locus of points equidistant from two lines is the angle bisector of the angle formed by those lines. 3. **Step-by-step solution:** - The problem asks for points inside ABCD equidistant from sides AB and BC. - Sides AB and BC meet at vertex B, forming an angle \(\angle ABC\). - The locus of points equidistant from AB and BC is the angle bisector of \(\angle ABC\). - Therefore, the locus is the ray starting at B that bisects \(\angle ABC\). 4. **Conclusion:** The locus of points inside quadrilateral ABCD equidistant from AB and BC is the angle bisector of \(\angle ABC\) starting at vertex B and extending inside the quadrilateral.