1. **State the problem:** We have triangle $ABC$ with $\angle CAB = 82^\circ$ and $\angle C = 68^\circ$. The perpendicular bisector of side $AB$ intersects $AB$ at $D$ and $BC$ at $E$. We need to find the measure of $\angle CAE$. 2. **Find the missing angle in triangle $ABC$:** The sum of angles in a triangle is $180^\circ$. So, $$\angle B = 180^\circ - \angle CAB - \angle C = 180^\circ - 82^\circ - 68^\circ = 30^\circ.$$ 3. **Understand the perpendicular bisector:** Since $D$ is the midpoint of $AB$ and the bisector is perpendicular to $AB$, $AD = DB$ and $DE \perp AB$. 4. **Analyze triangle $ABE$ and point $E$ on $BC$:** The perpendicular bisector of $AB$ intersects $BC$ at $E$. Because $D$ is midpoint of $AB$, $DE$ is perpendicular to $AB$. 5. **Use angle properties:** Since $DE$ is perpendicular to $AB$, $\angle ADE = 90^\circ$. Also, $AD = DB$. 6. **Find $\angle CAE$:** Note that $\angle CAE$ is the angle between $CA$ and $AE$. Since $E$ lies on $BC$, and $D$ is midpoint of $AB$, the line $DE$ is the perpendicular bisector of $AB$. 7. **Key insight:** Because $DE$ is the perpendicular bisector of $AB$, $AE$ is the reflection of $BE$ about $DE$. This implies $\angle CAE = \angle CAB - \angle B = 82^\circ - 30^\circ = 52^\circ$. **Final answer:** $$\boxed{52^\circ}$$