Question Video: Using the Properties of Parallelograms to Find an Unknown Length Mathematics

Video Transcript

In the given figure, what is the length of line segment 𝐹𝐸?

Let’s begin by considering the information shown in the figure.

We notice that we have two pairs of parallel line segments, with the first being that line segments 𝐴𝐷 and 𝐵𝐸 are parallel. The second pair of parallel line segments are 𝐴𝐵 and 𝐷𝐶. So, by definition, that means that the quadrilateral 𝐴𝐵𝐶𝐷 is a parallelogram, since these are quadrilaterals with both pairs of opposite sides parallel.

One of the properties of parallelograms that will be useful in this problem is that opposite sides are equal in measure or congruent. So, this line segment 𝐵𝐶, which is already marked as congruent to line segment 𝐸𝐶, is also congruent to the line segment 𝐴𝐷, which is opposite it in the parallelogram. So, we can mark it with the same two lines as the other two congruent line segments.

We are asked to find the length of the line segment 𝐹𝐸. It doesn’t appear as though we have enough information, as we only have the length of one line segment on the diagram. So, a good approach would be to check if perhaps the two triangles 𝐴𝐷𝐹 and 𝐶𝐸𝐹 have a mathematical relationship. For example, we can check if they are congruent triangles.

As we have the pair of parallel line segments 𝐴𝐷 and 𝐵𝐸, we can work out information about some of the angle measures in these triangles. Using these parallel line segments and the transversal 𝐴𝐸, we can note that angles 𝐷𝐴𝐹 and 𝐶𝐸𝐹 are congruent, as these are alternate angles. Similarly, angles 𝐴𝐷𝐹 and 𝐸𝐶𝐹 are also alternate angles and so are congruent.

Therefore, we have determined that we have two pairs of congruent angles. And we have already established that the pair of included sides, 𝐴𝐷 and 𝐸𝐶, between these angles are congruent. So, this proves that triangles 𝐴𝐷𝐹 and 𝐸𝐶𝐹 are congruent by using the ASA, or angle-side-angle, congruence criterion. This will allow us to find the length of line segment 𝐹𝐸.

In the congruent triangles, the side which is corresponding to line segment 𝐹𝐸 is line segment 𝐹𝐴. As these are corresponding sides, they are congruent. And the length of line segment 𝐹𝐴 is six centimeters.

So, we have found the answer of six centimeters for the length of line segment 𝐹𝐸 by first using the properties of parallelograms and then proving that there is a pair of congruent triangles.