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 $A B$ at point $B$ has measure $180 minus 139$ , or $41$ , degrees. It's given that segment $A B$ is parallel to segment $D E$ . Extending segment $B C$ at point $C$ and extending segment $D E$ 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 $A B$ is parallel to segment $D E$ , alternate interior angles are congruent. Angle $C X D$ and the exterior angle formed by extending segment $A B$ at point $B$ are alternate interior angles. Therefore, the measure of angle $C X D$ is $41$ degrees. It's given that the measure of angle $D$ in pentagon $A B C D E$ is $174$ degrees. Therefore, angle $C D X$ has measure $180 minus 174$ , or $6$ , degrees. Since angle $C$ in pentagon $A B C D E$ is an exterior angle of triangle $C D X$ , it follows that the measure of angle $C$ is the sum of the measures of angles $C D X$ and $C X D$ . Therefore, the measure, in degrees, of angle $C$ is $6 plus 41$ , or $47$ .

Alternate approach: A line can be created that's perpendicular to segments $A B$ and $D E$ and passes through point $C$ . Extending segments $A B$ and $D E$ 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 $B X C$ and triangle $D Y C$ . It's given that the measure of angle $B$ is $139$ degrees. Therefore, angle $C B X$ has measure $180 minus 139$ , or $41$ , degrees. Since the measure of angle $C B X$ is $41$ degrees and the measure of angle $B X C$ is $90$ degrees, it follows that the measure of angle $X C B$ is $180 - 90 - 41$ , or $49$ , degrees. It's given that the measure of angle $D$ is $174$ degrees. Therefore, angle $Y D C$ has measure $180 minus 174$ , or $6$ , degrees. Since the measure of angle $Y D C$ is $6$ degrees and the measure of angle $C Y D$ is $90$ degrees, it follows that the measure of angle $D C Y$ is $180 - 90 - 6$ , or $84$ , degrees. Since angles $X C B$ , $D C Y$ , and angle $C$ in pentagon $A B C D E$ form segment $X Y$ , 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 $47$ .