If 𝐸𝐵𝑀𝐿 is an isosceles trapezoid, then which of the following is true? Option (A) 𝑀𝐸 equals 𝑀𝐵, option (B) 𝐿𝐸 equals 𝐿𝐵, option (C) 𝐿𝑀 equals 𝐸𝐵, option (D) 𝑀𝐸 equals 𝐿𝐵, option (E) 𝐿𝐺 equals 𝐺𝐵.
Let’s 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, we’re given the information that 𝐸𝐵𝑀𝐿 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. Let’s see if this will help us answer the question.
In our diagram, the nonparallel sides would be 𝑀𝐵 and 𝐿𝐸. So we could write the statement that 𝐿𝐸 equals 𝑀𝐵. However, if we look at the answer options, we don’t have this included. We don’t even have anything similar, for example, 𝐸𝐿 equals 𝑀𝐵 or 𝐸𝐿 equals 𝐵𝑀 or anything like it. So let’s go back to the properties of an isosceles trapezoid and see if there’s 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 𝑀𝐸 would be equal to the other diagonal of 𝐿𝐵. We can see that this fits the answer given in option (D). It would also have been valid to write that 𝑀𝐸 equals 𝐵𝐿 or 𝐸𝑀 equals 𝐵𝐿. Any of these formats would indicate that the two diagonals are congruent.
Before we finish this question, let’s have a quick check of the other answer options. Option (A) says 𝑀𝐸, one of the diagonals, is equal to 𝑀𝐵, one of the legs of the trapezoid. This could be true potentially, but we can’t say it for sure, so option (A) isn’t true.
Option (B) is similar because it says that 𝐿𝐸, the leg of this trapezoid, is equal to the diagonal 𝐿𝐵. Once again, we don’t know this for sure. It’s not a property of an isosceles trapezoid.
Option (C) compares the two parallel sides and says that these would be equal in length: 𝐿𝑀 equals 𝐸𝐵. Well, if we had an isosceles trapezoid and we’re told that the two parallel sides are the same length, then we would in fact have more of a rectangle shape. Once again, this isn’t a particular property of trapezoids or isosceles trapezoids. So option (C) is incorrect.
Finally, option (E) compares 𝐿𝐺 and 𝐺𝐵. These are two portions of the diagonal 𝐿𝐵. We can see in this diagram that 𝐿𝐺 and 𝐺𝐵 are not the same length. And in fact, there’s 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: 𝑀𝐸 equals 𝐿𝐵.