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

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

In the figure, what is the length of line segment πΉπΈ?

03:15

### Video Transcript

In the given figure, what is the length of line segment πΉπΈ?

Letβ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 π΄π· and π΅πΈ are parallel. The second pair of parallel line segments are π΄π΅ and π·πΆ. So, by definition, that means that the quadrilateral π΄π΅πΆπ· 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 π΅πΆ, which is already marked as congruent to line segment πΈπΆ, is also congruent to the line segment π΄π·, 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 πΉπΈ. It doesnβ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 π΄π·πΉ and πΆπΈπΉ have a mathematical relationship. For example, we can check if they are congruent triangles.

As we have the pair of parallel line segments π΄π· and π΅πΈ, we can work out information about some of the angle measures in these triangles. Using these parallel line segments and the transversal π΄πΈ, we can note that angles π·π΄πΉ and πΆπΈπΉ are congruent, as these are alternate angles. Similarly, angles π΄π·πΉ and πΈπΆπΉ 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, π΄π· and πΈπΆ, between these angles are congruent. So, this proves that triangles π΄π·πΉ and πΈπΆπΉ are congruent by using the ASA, or angle-side-angle, congruence criterion. This will allow us to find the length of line segment πΉπΈ.

In the congruent triangles, the side which is corresponding to line segment πΉπΈ is line segment πΉπ΄. As these are corresponding sides, they are congruent. And the length of line segment πΉπ΄ is six centimeters.

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