Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.

Explanation

Given Two lines m and n are parallal and another two lines p and q are respectively perpendicular to m and n.
i.e., `" " p bot m,p bot n,q bot m,q bot`n
To prove `p|""|q`
Proof Since, m`|""|`n and p is perpendicular to m and n .

`:." "/_1 = /_10 = 90^(@)" " `[corresponding angles]
Simillarly ,`" "/_2 = /_9 = 90^(@)" "`[corresponding angles] [`:'pbotm and pbotn`]
`:." "/_4 = /_9 = 90^(@) and /_3 = /_10 = 90^(@) " "`[alternate interior angles]
Similarly, if m`|""|`n and q is perpendicular to m and n .
Then`" "/_7 = 90^(@)and /_11 = 90^(@)`
Now,`" "/_3 +/_7 = /_90^(@) +/_90^(@) = /_180^(@)`
So, sum of two interior angles is supplementary.
we know that, if a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.
Hence, p`|""|`q.