Question

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

Answer 1

Given: A line segment AB of length 4 cm.

To construct: To draw a line perpendicular to AB through A and B, respectively.

Use the following steps of construction.

1. Draw AB = 4 cm.

2. With 4 as centre and radius more than `1/2` AB, draw an arc that is intersect AB at E.

3. With E as centre and with same radius as above draw an arc which intersect previous arc at F.

4. Again, taking F as centre and with same radius as above draw an arc which intersect previous arc (obtained in step ii) at G.

5. With G and F are centres, draw arcs which intersect each other at H.

6. Join AH. So, AX is perpendicular to AB at A. Similarly, draw BY ⊥ AB at B.

Now, we know that if two lines are parallel, then the angle between them will be 0° or 180°. 

So, ∠XAB = 90°  ...[XA ⊥ AB]

And ∠YBA = 90° ...[YB ⊥ AB]

∠XAB + ∠YBA = 90° + 90° = 180°

Hence, the lines XA and YS are parallel.  ...[It sum of interior angle on same side of transversal is 180°, then the two lines are parallel]