6.1.3 Conditions for Parallelograms Key Objectives • Prove that a given quadrilateral is a parallelogram.
Theorems, Postulates, Corollaries, and Properties • Theorem If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram. • Theorem If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. • Theorem If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. • Theorem If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. • Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Example 1 Verifying Figures are Parallelograms
This is a condition of a quadrilateral being a parallelogram. If both pairs of opposite sides of the quadrilateral are congruent, then it is a parallelogram. Note the symbols for congruent sides in the figure. This is a condition of a quadrilateral being a parallelogram. If both pairs of the opposite angles of a quadrilateral are congruent, then that quadrilateral is a parallelogram. Note the symbols for congruent angles in the figure.
This is a condition of a quadrilateral being a parallelogram. If an angle of a quadrilateral is supplementary to both of its consecutive angles, then that quadrilateral is a parallelogram.
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