l, m and n are three parallel lines intersected by transversals p and q such that l, m and n cut off equal intercepts AB and BC on p (see Fig. 8.28). Show that l, m and n cut off equal intercepts DE and EF on q also

Explanation

Given:`AB=BC`
To prove:`DE=EF`
Construction: Join AF intersecting m at `G`.
Now the trapezium `ACFD` is divided into two triangles: `ΔACF and ΔAFD.`
In `ΔACF, B` is the mid point of `AC` and `BG || CF` (as m || n )
∴ By mid point theorem, `G` is the mid point of `AF`.
In `ΔAFD, G` is the mid point of `AF and GE || AD` (as l || m)
⇒` E` is the mid point of `DF` (by mid point theorem)
⇒ `therefore DE = EF`