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Answer :
The question involves the intercept theorem, which states that if three or more parallel lines cut equal intercepts on a transversal, then they will also cut equal intercepts on any other transversal.
In this situation, we have parallel lines cutting a line at points [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] such that the distance between [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is equal to the distance between [tex]\( B \)[/tex] to [tex]\( C \)[/tex], i.e., [tex]\(|AB| = |BC|\)[/tex]. These parallel lines cut another transversal at points [tex]\( X, Y, \)[/tex] and [tex]\( Z \)[/tex].
According to the intercept theorem, this set of parallel lines will cut equal intercepts on the second transversal as well. Therefore, the segments [tex]\( |XY| \)[/tex] and [tex]\( |YZ| \)[/tex] will also be equal:
[tex]\[
|XY| = |YZ|
\][/tex]
This means each segment on the second transversal between the respective intersections will be equal, maintaining the proportionality defined by the parallel lines.
This theorem significantly helps in various geometric problems where constructions involving parallel lines and transversals are involved, ensuring that proportional segments are a characteristic feature in these figures.
In this situation, we have parallel lines cutting a line at points [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] such that the distance between [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is equal to the distance between [tex]\( B \)[/tex] to [tex]\( C \)[/tex], i.e., [tex]\(|AB| = |BC|\)[/tex]. These parallel lines cut another transversal at points [tex]\( X, Y, \)[/tex] and [tex]\( Z \)[/tex].
According to the intercept theorem, this set of parallel lines will cut equal intercepts on the second transversal as well. Therefore, the segments [tex]\( |XY| \)[/tex] and [tex]\( |YZ| \)[/tex] will also be equal:
[tex]\[
|XY| = |YZ|
\][/tex]
This means each segment on the second transversal between the respective intersections will be equal, maintaining the proportionality defined by the parallel lines.
This theorem significantly helps in various geometric problems where constructions involving parallel lines and transversals are involved, ensuring that proportional segments are a characteristic feature in these figures.
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