Question

While constructing a parallel line to a given line, we
[a] copy a segment
[b] bisect a segment
[c] Copy an angle
[d] Construct a perpendicular

Answer 1

Hint: Recall the definition of parallel lines and recall the various properties parallel lines satisfy with the transversal. Hence find which of the options is a valid method for constructing parallel lines. Hence find which of the options is correct.

Complete step-by-step answer:
Before solving the question, we need to understand the definition of parallel lines and the various properties parallel lines satisfy with a transversal.
Parallel Lines: Two lines l and m are said to be parallel to each other if they never intersect each other (or in other words intersect at infinity).
Consider two lines l and m and a transversal t as shown.

If l and m are parallel, then the following properties are satisfied
[1] Pair of corresponding angles are equal
[2] Pair of alternate interior angles are equal
[3] Consecutive pair of interior angles are supplementary
[4] Alternate exterior angles are equal
Also, if any of the four properties is satisfied then the lines are parallel to each other.
Hence, we create parallel lines by making corresponding angles equal or making co-interior angles supplementary (each 90).
Method 1: Using the fact that the corresponding angles are equal.
Consider a line l as shown and we want to construct another line parallel to it passing through point P as shown in the diagram below

Steps of construction:
[1] Mark any convenient point Q on l.

[2] Join PQ. Mark points G on L and H on PQ as shown

[3] Construct angle CPH such that $\angle CPH=\angle GQP$ (Copy the angle)

[5] Extend line CP on both sides. CP is the line passing through P and parallel to l.

Method 2:
Suppose l is a line and P is a given point and we need to construct a parallel line to l through P

Steps of construction:
[1] Draw perpendicular PQ on line l.

[2] Draw angle $CPQ=90{}^\circ $ and extend CP on both sides

[3] Hence CP is the required parallel line through P.
Hence options C and D are correct.

Note: Justification of the constructions:
Justification of Method 1:

By construction, we have
$\angle GQP=\angle CPH$
But these are corresponding angles. Hence by the converse of corresponding angle axiom, we have CP||GQ.
Hence proved
Justification of Method 2:

$\angle P+\angle Q=180{}^\circ $
But these are co-interior angles. Hence CP||l
Hence proved.