# Question Video: Triangles Sharing the Same Base and between Parallel Lines Mathematics • 6th Grade

If the areas of ΞπΏππ΄ and Ξππ΄πΊ are the same, which of the following must be true? [A] ππΏ = ππΊ [B] Line segment ππΏ β₯ line segment ππΊ [C] π΄π = π΄πΊ [D] ππΊ = ππΏ [E] line segment ππΊ β₯ line segment ππΏ

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### Video Transcript

If the areas of triangle πΏππ΄ and triangle ππ΄πΊ are the same, which of the following must be true? Is it (A) ππΏ equals ππΊ? (B) Line segment ππΏ is parallel to line segment ππΊ. Is it (C) π΄π equals π΄πΊ? (D) ππΊ equals ππΏ. Or (E) line segment ππΊ is parallel to line segment ππΏ.

Now, be a little bit careful. Each of these lines is drawn on a two-dimensional plane. This is not a 3D shape. It does look a little bit like a square-based pyramid, but itβs certainly not. Weβre told that the areas of triangle πΏππ΄ and triangle ππ΄πΊ are the same. Letβs highlight these in pink and yellow, respectively. This is triangle πΏππ΄, and we have triangle ππ΄πΊ here.

And so letβs recall what we know about the equality of the areas of two triangles. We know that the areas of two triangles will be equal if these triangles share the same base and their vertices lie on a straight line parallel to the base, or if the triangles share the same vertex and have bases of equal length along the same line. And so what weβre going to do to be able to use one of these definitions is split our triangles up.

If we split triangle πΏππ΄ and ππ΄πΊ as shown by adding the line segments π·π and πΊπΈ, respectively, then we can begin by considering a shared vertex π΄. This corresponds to the second definition. These triangles share the same vertex. And so weβll be able to say that the area of triangle π΄πΊπΈ will be equal to the area of triangle π΄π·π if their bases are equal length and they lie along the same line. Well, in fact, the dashes on line segment πΊπΈ and π·π indicate that theyβre the same length. And we can quite clearly see from the picture that they do lie along the same line. This must mean that the area of triangle π΄πΊπΈ is equal to the area of triangle π΄π·π.

Well, since the areas of our larger triangles are the same, this in turn means that the area of triangle πΊπΈπ must be equal to the area of triangle π·ππΏ. These triangles donβt have a shared vertex. But they do have bases of equal length, which lie on the same line. Thatβs πΊπΈ and π·π. And so we can use our first definition.

Now, it doesnβt really matter that πΊπΈ and π·π are different line segments. The simple fact that they lie on the same line and theyβre equal in length is good enough as saying that they share the same base. And so we can say that for triangle πΊπΈπ to have an equal area to triangle π·ππΏ, the vertices π and πΏ β thatβs these two β must lie on a line parallel to the base. And so the lines passing through πΊπΈπ·π and ππΏ must be parallel. If we go to our options, we see that corresponds to option (D). The line segment ππΏ must be parallel to the line segment ππΊ. The answer is (B).