What is the intersection of the plane through 𝐴𝐵𝐵 dash 𝐴 dash and the plane through 𝐵𝐶𝐶 dash 𝐵 dash?
Let’s first consider the plane 𝐴𝐵𝐵 dash 𝐴 dash as shown in the diagram. Well this plane has four edges 𝐴𝐵, 𝐵𝐵 dash, 𝐵 dash 𝐴 dash, and 𝐴 dash 𝐴 and four vertices: 𝐴, 𝐵, 𝐵 dash, and 𝐴 dash.
Now let’s consider the second plane, 𝐵𝐶𝐶 dash 𝐵 dash. This plane also has four edges 𝐵𝐶, 𝐶𝐶 dash, 𝐶 dash 𝐵 dash, and 𝐵 dash 𝐵. It also has four vertices: 𝐵, 𝐶, 𝐶 dash, and 𝐵 dash.
Where do these two planes intersect? Do they intersect at a vertex or point or do they intersect at an edge? Well it is clear from the diagram that the two planes intersect at 𝐵𝐵 dash.
Therefore, the intersection of the plane through 𝐴𝐵𝐵 dash 𝐴 dash and the plane through 𝐵𝐶𝐶 dash 𝐵 dash is the line or edge 𝐵𝐵 dash. You will notice that when looking at the original two planes, the two vertices that appear in both of them are 𝐵 and 𝐵 dash.
This means that the planes must intersect at 𝐵 and 𝐵 dash. We could write this as the intersection of 𝐴𝐵𝐵 dash 𝐴 dash and 𝐵𝐶𝐶 dash 𝐵 dash is 𝐵𝐵 dash.