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The correct answer is $47$. It's given that the measure of angle $B$ is $139$ degrees. Therefore, the exterior angle formed by extending segment $AB$ at point $B$ has measure $180-139$, or $41$, degrees. It's given that segment $AB$ is parallel to segment $DE$. Extending segment $BC$ at point $C$ and extending segment $DE$ at point $D$ until the two segments intersect results in a transversal that intersects two parallel line segments. One of these intersection points is point $B$, and let the other intersection point be point $X$. Since segment $AB$ is parallel to segment $DE$, alternate interior angles are congruent. Angle $CXD$ and the exterior angle formed by extending segment $AB$ at point $B$ are alternate interior angles. Therefore, the measure of angle $CXD$ is $41$ degrees. It's given that the measure of angle $D$ in pentagon $ABCDE$ is $174$ degrees. Therefore, angle $CDX$ has measure $180-174$, or $6$, degrees. Since angle $C$ in pentagon $ABCDE$ is an exterior angle of triangle $CDX$, it follows that the measure of angle $C$ is the sum of the measures of angles $CDX$ and $CXD$. Therefore, the measure, in degrees, of angle $C$ is $6+41$, or $47$.

Alternate approach: A line can be created that's perpendicular to segments $AB$ and $DE$ and passes through point $C$. Extending segments $AB$ and $DE$ at points $B$ and $D$, respectively, until they intersect this line yields two right triangles. Let these intersection points be point $X$ and point $Y$, and the two right triangles be triangle $BXC$ and triangle $DYC$. It's given that the measure of angle $B$ is $139$ degrees. Therefore, angle $CBX$ has measure $180-139$, or $41$, degrees. Since the measure of angle $CBX$ is $41$ degrees and the measure of angle $BXC$ is $90$ degrees, it follows that the measure of angle $XCB$ is $180-90-41$, or $49$, degrees. It's given that the measure of angle $D$ is $174$ degrees. Therefore, angle $YDC$ has measure $180-174$, or $6$, degrees. Since the measure of angle $YDC$ is $6$ degrees and the measure of angle $CYD$ is $90$ degrees, it follows that the measure of angle $DCY$ is $180-90-6$, or $84$, degrees. Since angles $XCB$, $DCY$, and angle $C$ in pentagon $ABCDE$ form segment $XY$, it follows that the sum of the measures of those angles is $180$ degrees. Therefore, the measure, in degrees, of angle $C$ is $180-49-84$, or $47$.