Can you draw a pair of parallel lines using a ruler and a set square? Fig. 5.21 shows how you can do it. Draw a line l with a scale. By sliding your set square you can make two lines perpendicular to line l.
Answer
Yes. Here is the method shown in Fig. 5.21.
Draw line l with the scale.
Place one edge of the set square along l and draw a line along the edge that stands at 90° to it.
Slide the set square along l to a new place and draw a second line along that same 90° edge.
Both new lines make 90° with l These 90° angles are corresponding angles Equal corresponding angles ⟹ the two new lines are parallel
Why it happens: Sliding the set square does not turn it, so the drawing edge keeps pointing the same way. Both new lines therefore lean identically and can never meet.
Q2.
Are these two lines parallel to each other? How are we sure that they are parallel to each other? What angles are formed between these lines and line l?
Answer
Yes, they are parallel. The angles between each of them and line l are 90°.
Angle of first line with l = 90° (set square) Angle of second line with l = 90° Treat l as a transversal of the two new lines Corresponding angles = 90° = 90° → equal ⟹ the two lines are parallel
Why it happens: We are not relying on how the picture looks. We are using the rule that equal corresponding angles guarantee parallelism — reasoning, not eyesight.
Did you know? A useful consequence: two lines that are both perpendicular to the same line are parallel to each other.
Q3.
Draw two more parallel lines using the long side of the set square as shown in Fig. 5.22.
Answer
This time the drawing edge is not at 90° to l, but the idea is exactly the same.
Keep the scale fixed on the paper as a guide rail.
Hold the long side of the set square against the scale and draw a line along the opposite edge.
Slide the set square along the scale — without turning it — and draw a second line along the same edge.
Both lines make the same angle with the scale's edge That edge acts as a transversal Corresponding angles equal ⟹ the lines are parallel
Why it happens: The scale stops the set square from rotating. Only sliding is allowed, and sliding preserves direction.
Q4.
How do you know these two lines are parallel? Can you check if the corresponding angles are equal?
Answer
Draw any line cutting both of them and compare the two angles that sit in the same position.
By protractor: measure the angle at each crossing on the same side of the transversal and in the same position. The two readings agree.
By tracing paper: trace one angle, slide the tracing along the transversal to the other crossing. It fits exactly.
If the two readings are equal → the lines are parallel If they differ by even 2° or 3° → the lines are not parallel and they will meet somewhere far away
Why it happens: Equality of corresponding angles is both necessary and sufficient for a pair of lines to be parallel. So this single check settles the question completely.
Tip: Small measuring errors are normal. If your two readings differ by less than a degree, treat the lines as parallel and blame the protractor and the thickness of the pencil line.