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Tests for Parallelograms Advanced Geometry Polygons Lesson 3

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If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. There are 5 ways to prove that a quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Definition of parallelogram

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Example: Determine whether each quadrilateral is a parallelogram. Justify your answer. Parallelogram Each pair of opposite angles is congruent.

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Example: Write a proof of the statement: If a diagonal of a quadrilateral divides the quadrilateral into two congruent triangles, then the quadrilateral is a parallelogram.

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Example: Determine whether a figure with the given vertices is a parallelogram. Use the method indicated. A(-3, 0), B(-1, 3), C(3, 2), D(1, -1); Slope Formula

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Method: Parallelogram Test: Slope Formula Distance Formula Distance & Slope Formulas Midpoint Formula Def. of parallelogram -Test to see if opposite sides are parallel. Opposite sides are congruent. -Find the length of each side to make sure opposite sides have the same lengths. One pair of opposite sides are both parallel and congruent. -Find the lengths and slopes of the same pair of opposite sides. Diagonals bisect each other. -Find the midpoint of each diagonal to make sure it is the same point.

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Example: Determine whether a figure with the given vertices is a parallelogram. Use the method indicated. F(-2, 4), G(4, 2), H(4, -2), F(-2, -1); Distance and Slope Formulas

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