What is the measure of one of the larger angles of the...

What is the measure of one of the larger angles of the parallelogram that has vertices at $(-2,-2),$ $(0,1),(5,1),$ and $(3,-2) ?$
(A) $117.2^{\circ}$
(B) $123.7^{\circ}$
(C) $124.9^{\circ}$
(D) $125.3^{\circ}$
(E) $131.0^{\circ}$

Key Concepts

Supplementary Angles

Supplementary angles are two angles whose measures add up to 180°. This characteristic is directly applicable to parallelograms, where any two adjacent angles are supplementary. Determining one angle through the use of vectors or other methods automatically allows for the computation of its supplementary counterpart.

Coordinate Geometry

Coordinate geometry, or analytic geometry, utilizes algebraic methods to solve geometric problems by representing figures in a coordinate system. This approach allows one to calculate distances, slopes, and angles using numerical coordinates, thereby simplifying the process of analyzing geometric shapes such as parallelograms.

Parallelogram Properties

A parallelogram is a quadrilateral where opposite sides are both parallel and equal in length. One of its key properties is that opposite angles are equal and adjacent angles are supplementary, meaning they add up to 180°. This property is crucial in determining unknown angles once one of the angles has been computed.

Vector Dot Product and Angle Calculation

The dot product is a fundamental tool in vector arithmetic that relates two vectors and helps compute the angle between them through the formula v · w = |v| |w| cos?. This method is particularly useful in coordinate geometry when the sides of a figure are represented as vectors, allowing for precise calculation of the angles formed between them.

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