6.1.3 Conditions for Parallelograms (continued)
Quadrilateral figures are determined to be parallelograms or not.
It is given in the first figure that one of the pairs of opposite sides of the quadrilateral is both congruent and parallel. Therefore, this is a parallelogram by a condition for parallelograms. It is given in the second figure that one pair of opposite sides is congruent and one pair of consecutive angles is congruent. This is not enough information to determine that the quadrilateral is a parallelogram. Remember not to make any assumptions based on what the figure looks like. For example, if the congruent angles are not right angles, then the congruent sides are not parallel.
Example 3 Proving Parallelograms in the Coordinate Plane
A quadrilateral in the coordinate plane is proven here to be a parallelogram. The coordinates of the
vertices of the parallelogram are given. Calculate the slopes of the sides of the quadrilateral. For example the slope of AB = ( y B − y A )/( x B − x A ) = (1 − 3)/(1 − ( − 3)) = − 2/4 = − 1/2.
Calculating all slopes shows that AB || DC and AD || BC . Since both pairs of opposite sides are parallel, by the conditions for parallelograms, the quadrilateral ABCD is a parallelogram. A quadrilateral in the coordinate plane is proven here to be a parallelogram. The coordinates of the vertices of the parallelogram are given. Calculate the slopes of two of the sides of the quadrilateral. For example, the slope of FG = ( y G − y F )/( x G − x F ) = (5 − 5)/(1 − ( − 3)) = 0/4 = 0. The slope of JH is also 0. Therefore, FG || JH . Calculate the lengths FG and JH . For example, the length FG is (1 − ( − 3)) = 4. This is also the length JH . Since the opposite sides are congruent and parallel, by a condition for parallelograms, the quadrilateral FGHJ is a parallelogram.
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