Is the given quadrilateral a parallelogram?
So we’ve been given a diagram of a quadrilateral 𝐹𝐺𝐻𝐽 and asked to determine whether it’s a parallelogram. We’ve been given some information about some lengths in this quadrilateral and also two of the angles. In order to determine whether the quadrilateral is a parallelogram, we need to consider the properties of lengths and angles in a parallelogram.
Firstly, a key fact about the lengths of sides in a parallelogram. Opposite sides are congruent. This means that, in our quadrilateral, 𝐹𝐺 would need to be the same length as 𝐽𝐻 and 𝐻𝐺 would need to be the same length as 𝐹𝐽.
We can see that the lengths we’ve been given for one pair of opposite sides are not the same. 𝐹𝐺 is 45 inches, whereas 𝐽𝐻 is 44 inches. Therefore, based on this condition alone, the quadrilateral cannot be a parallelogram.
Now this is enough to be able to answer the question. But we will just mention the angles as well. We’ve been given a pair of consecutive interior angles in the quadrilateral: 96 degrees and 84 degrees. Now these angles do in fact fulfill one of the properties of angles in a parallelogram, which is that consecutive angles are supplementary. They add to 180 degrees.
However, we haven’t been told about the other angles in the quadrilateral. So we can’t conclude that this relationship is true for all pairs of consecutive angles. So was this quadrilateral may at least have something in common with a parallelogram? Showing that the opposite sides 𝐹𝐺 and 𝐺𝐻 are not congruent is enough to conclude that it isn’t a parallelogram.