### Video Transcript

Find the length of line segment πΈπΆ.

Before we start looking at the length of πΈπΆ, letβs have a look at our diagram and see if we can work out any of the shapes in it. If we look at our shape π·π΄π΅πΆ, we can see that there are two parallel sides and two 90-degree angles. So the angle π·π΄π΅ and the angle π·πΆπ΅ must also be 90 degrees, as we would have pairs of supplementary angles. We can conclude, therefore, that our shape π·π΄πΆπ΅ is a rectangle. We have been asked to find the length of line segment πΈπΆ. But so far, using the fact that π·π΄πΆπ΅ is a rectangle isnβt enough information to help us with this. Letβs then have a look at the other shapes in this diagram.

We have a triangle πΉπ·π΄ and a triangle πΉπΆπΈ. Letβs start by looking at our side length π·πΉ in our purple Triangle and our side length πΉπΆ in our orange triangle. We can see from our diagram that these two lengths are congruent. That is, line segment π·πΉ is equal to the line segment πΉπΆ. We can see that the measure of angle πΉπ·π΄ is 90 degrees. And if we look at the angle πΉπΆπΈ in triangle πΉπΆπΈ, then we can note that this must be 90 degrees since we have a straight line with one 90-degree angle on it. So we can say that the measure of angle πΉπΆπΈ is 90 degrees and giving us that we have two equivalent angles in our triangles.

Next, if we have a look at the angle π·πΉπ΄ in triangle πΉπ·π΄, then we can say that it must be equal to the measure of angle πΈπΉπΆ in triangle πΉπΆπΈ. Since these are opposite angles. So if we look at the information weβve just recorded, we can see that we have an equivalent angle and equivalent side and another equivalent angle. This means that we can say that our triangles πΉπ·π΄ and πΉπΆπΈ are congruent using the angle side angle congruency criterion.

Letβs see if this helps us workout the length of line segment πΈπΆ. We can establish since our triangles are congruent, then our length πΈπΆ in triangle πΉπΆπΈ is congruent to our length π·π΄ in triangle πΉπ·π΄. But what is the length of π·π΄? Well, if we return to the fact that our shape π·π΄πΆπ΅ is a rectangle, we know that opposite sides are equal in length. So the length of π·π΄ must be equal to the length of πΆπ΅, which is 7.5 centimetres. And so we conclude that because π·π΄ is equal to 7.5 centimetres, then the length of our line segment πΈπΆ is also 7.5 centimetres.