In convex pentagon A B C D E , segment A B is parallel to segment D E . The

Question

In convex pentagon A B C D E , segment A B is parallel to segment D E . The measure of angle B is 139 degrees, and the measure of angle D is 174 degrees. What is the measure, in degrees, of angle C ?

Accepted Answer

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 - 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 - 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 + 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 - 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 - 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 .

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