Video Transcript
True or False: In the given figure,
if 𝐴𝐵 equals 𝐵𝐶 equals 𝐶𝐷, then 𝑋𝑌 equals 𝑌𝑍 equals 𝑍𝐿.
In the figure, we can observe that
we have three congruent line segments marked, which comes from the fact that we are
told that these three line segments 𝐴𝐵, 𝐵𝐶, and 𝐶𝐷 are all equal in
length. We can also see from the markings
on the diagram that there are four parallel lines: the lines 𝐴𝑋, 𝐵𝑌, 𝐶𝑍, and
𝐷𝐿. And the fact that we have parallel
lines being split by a transversal line 𝐴𝐷 means that we can apply a very
important property.
If a set of parallel lines divide a
transversal into segments of equal length, then they divide any other transversal
into segments of equal length. So, this tells us that because
𝐴𝐵, 𝐵𝐶, and 𝐶𝐷 are all of equal length, then the line segments 𝑋𝑌, 𝑌𝑍, and
𝑍𝐿 are all of equal length.
But we must be careful of
misunderstanding this property. For example, if we said that the
lengths of the line segments on the left transversal are all 𝑑 length units, it
would not mean that the lengths of the line segments on the right transversal are
also 𝑑 units. But instead, we just know that all
the line segments on the transversal line 𝑋𝑍 are all the same length as each
other, for example, 𝑚 length units.
We can write that 𝑋𝑌 equals 𝑌𝑍
equals 𝑍𝐿. And therefore, the statement in the
question is true.
