Question

Question 2
Draw a line segment AB of 4cm in length. Draw a line perpendicular to AB through A and B, respectively. Are these lines parallel?

Answer 1

To draw a line perpendicular to AB through A and B, respectively, use the following steps of construction.
(i) Draw a line segment AB = 4cm.
(ii) Taking A as the centre and radius more than $\frac{1}{2}$ AB(i.e., 2cm) draw an arc which intersects AB at E.
(iii) Taking E as the centre and with the same radius as above draw an arc which intersects the previous arc at F.
(iv) Again, taking F as the centre and with the same radius as above draw an arc which intersects previous arc( obtained in step (ii)) at G.

(v) Taking G and F are centres, draw arcs which intersect each other at H.
(vi) Join AH, thus AX is the perpendicular to AB at A. similarly, draw BY $\perp$ AB at B.
Now, we know that if two lines are parallel, then the angle between them will be $0^{\circ }or{180}^{\circ }$.
$Here,\angle XAB={90}^{\circ }[\because XA\perp AB]$
$And\angle YBA={90}^{\circ }[\because YB\perp AB]$
$\therefore \angle XAB+\angle YBA={90}^{\circ }+{90}^{\circ }={180}^{\circ }$
So, the lines XA and YB are parallel.
[ since, if the sum of interior angles on same side of transversal is ${180}^{\circ }$, then the two lines are parallel.]