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.