### Video Transcript

If πΈπ΅ππΏ is an isosceles trapezoid, then which of the following is true? Option (A) ππΈ equals ππ΅, option (B) πΏπΈ equals πΏπ΅, option (C) πΏπ equals πΈπ΅, option (D) ππΈ equals πΏπ΅, option (E) πΏπΊ equals πΊπ΅.

Letβs begin by reminding ourselves that a trapezoid is a quadrilateral with one pair of parallel sides, which we can see indicated in the diagram. Furthermore, weβre given the information that πΈπ΅ππΏ is an isosceles trapezoid. Isosceles trapezoids still have all the properties of trapezoids, so it still has one pair of parallel sides. But importantly here, nonparallel sides are congruent or the same length. Letβs see if this will help us answer the question.

In our diagram, the nonparallel sides would be ππ΅ and πΏπΈ. So we could write the statement that πΏπΈ equals ππ΅. However, if we look at the answer options, we donβt have this included. We donβt even have anything similar, for example, πΈπΏ equals ππ΅ or πΈπΏ equals π΅π or anything like it. So letβs go back to the properties of an isosceles trapezoid and see if thereβs anything else in an isosceles trapezoid that would be equal.

Well, in fact, there is one other thing. Because we have two nonparallel sides congruent, then this will mean that the diagonals of an isosceles trapezoid are also congruent. So, in our diagram, the diagonal ππΈ would be equal to the other diagonal of πΏπ΅. We can see that this fits the answer given in option (D). It would also have been valid to write that ππΈ equals π΅πΏ or πΈπ equals π΅πΏ. Any of these formats would indicate that the two diagonals are congruent.

Before we finish this question, letβs have a quick check of the other answer options. Option (A) says ππΈ, one of the diagonals, is equal to ππ΅, one of the legs of the trapezoid. This could be true potentially, but we canβt say it for sure, so option (A) isnβt true.

Option (B) is similar because it says that πΏπΈ, the leg of this trapezoid, is equal to the diagonal πΏπ΅. Once again, we donβt know this for sure. Itβs not a property of an isosceles trapezoid.

Option (C) compares the two parallel sides and says that these would be equal in length: πΏπ equals πΈπ΅. Well, if we had an isosceles trapezoid and weβre told that the two parallel sides are the same length, then we would in fact have more of a rectangle shape. Once again, this isnβt a particular property of trapezoids or isosceles trapezoids. So option (C) is incorrect.

Finally, option (E) compares πΏπΊ and πΊπ΅. These are two portions of the diagonal πΏπ΅. We can see in this diagram that πΏπΊ and πΊπ΅ are not the same length. And in fact, thereβs no property of isosceles trapezoids where the diagonals are bisected. And so option (E) is also incorrect, leaving us with the answer that, in an isosceles trapezoid, the diagonals are congruent: ππΈ equals πΏπ΅.