What is the measure of one of the larger angles of a parallelogram in the that has vertices with coordinates , , and ?

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The correct answer is C.

First, note that the angle of the parallelogram with vertex is one of the two larger angles of the parallelogram: Looking at the graph of the parallelogram in the makes this apparent. Alternatively, the sides of the angle of the parallelogram with vertex are a horizontal line segment with endpoints and and a line segment of positive slope with endpoints and that intersects the horizontal line segment at its left endpoint , so the angle must measure more than Since the sum of the measures of the four angles of a parallelogram equals , the angle with vertex must be one of the larger angles.

One way to determine the measure of the angle of the parallelogram with vertex is to apply the law of cosines to the triangle with vertices , , and . The length of the two sides of the angle with vertex are and ; the length of the side opposite the angle is . Let represent the angle with vertex and apply the Law of Cosines: , so . Therefore, the measure of one of the larger angles of the parallelogram is .

Another way to determine the measure of the angle of the parallelogram with vertex is to consider the triangle , , and . The measure of the angle of this triangle with vertex is less than the measure of the angle of the parallelogram with vertex . The angle of the triangle has opposite side of length and adjacent side of length , so the measure of this angle is . Therefore, the measure of the angle of the parallelogram with vertex is .

Yet another way to determine the measure of the angle of the parallelogram with vertex is to use trigonometric relationships to find the measure of one of the smaller angles, and then use the fact that each pair of a larger and smaller angle is a pair of supplementary angles. Consider the angle of the parallelogram with vertex ; this angle coincides with the angle at vertex of the right triangle with vertices at , , and , with opposite side of length and adjacent side of length , so the measure of this angle is . This angle, together with the angle of the parallelogram with vertex , form a pair of interior angles on the same side of a transversal that intersects parallel lines, so the sum of the measures of the pair of angles equals . Therefore, the measure of the angle of the parallelogram with vertex is .