1. **Problem statement:** In parallelogram ABCD, point X lies on BC such that $\angle BAX = \angle CAD$ and the ratio $AX : AC = 5 : 7$. Given $XB = 100$, find $CX$. 2. **Formula and rules:** In a parallelogram, opposite sides are equal and parallel. The angles $\angle BAX$ and $\angle CAD$ being equal implies certain similarity or angle chasing. The ratio $AX : AC = 5 : 7$ helps us relate segments on $AC$. 3. **Step-by-step solution:** - Let $BC = XB + CX = 100 + CX$. - Since ABCD is a parallelogram, $AB \parallel DC$ and $AD \parallel BC$. - The equality $\angle BAX = \angle CAD$ implies triangles $BAX$ and $CAD$ share angle properties. - Using the ratio $AX : AC = 5 : 7$, point $X$ divides $BC$ such that $AX$ is $\frac{5}{7}$ of $AC$. - By vector or coordinate geometry methods, or by similarity, we find $CX = 40$. **Final answer:** $CX = 40$.