1. **Problem Statement:** We have a cyclic quadrilateral inscribed in a circle with two pairs of equal sides. One interior angle at the bottom right corner is $38^\circ$, and an exterior angle adjacent to the top-right vertex is labeled $e^\circ$. We need to find the value of $e$. 2. **Key Properties:** - In a cyclic quadrilateral, opposite angles sum to $180^\circ$. - Exterior angle of a polygon equals the sum of the two opposite interior angles. - Equal sides imply isosceles triangles, which help find unknown angles. 3. **Step-by-step Solution:** - Let the quadrilateral be $ABCD$ with $D$ at the bottom right corner where the angle is $38^\circ$. - Since $ABCD$ is cyclic, $\angle D + \angle B = 180^\circ$. - Given $\angle D = 38^\circ$, so $\angle B = 180^\circ - 38^\circ = 142^\circ$. - The exterior angle $e$ at vertex $B$ is adjacent to $\angle B$ and equals $180^\circ - \angle B$ (linear pair). - Therefore, $e = 180^\circ - 142^\circ = 38^\circ$. 4. **Conclusion:** The exterior angle $e$ is equal to $38^\circ$. This matches the interior angle at $D$ due to the cyclic quadrilateral properties and the given equal sides.