Activity 4: Fig. 5.19 has a pair of parallel lines l and m (what is the notation used in the figure to indicate they are parallel?). Line t is the transversal across these two lines.
Answer
The notation is a pair of matching arrow heads (>) — one drawn on line l and one on line m.
Why it happens: A picture cannot be extended for ever, so we cannot see that two lines never meet. The arrow marks are a promise from the person who drew the figure that those lines are parallel.
Tip: If a figure has a second set of parallel lines, that set is marked with double arrow heads (>>), a third set with triple arrow heads, and so on. Perpendicular lines are marked with a small square instead.
Q2.
Take a tracing paper and trace ∠a on it. Now place this tracing paper over ∠b and see if the angles align exactly. Check the other corresponding angles in the figure using a protractor. Are all the corresponding angles equal to each other?
Answer
Yes. The tracing of ∠a falls exactly on ∠b, and every other corresponding pair matches too.
Corresponding pair
What you find
∠a and ∠b
equal
the other three pairs at the two crossings
equal as well
l ∥ m and t is a transversal ⟹ every pair of corresponding angles is equal
Why it happens: Sliding the tracing paper straight along the transversal from the first crossing to the second is a translation. Because l and m point in the same direction, the whole first crossing slides exactly onto the second — so every angle lands on its corresponding partner.
Check it yourself: Once one corresponding pair is known to be equal, the other three follow at once, using linear pairs and vertically opposite angles.
Q3.
Activity 5: In Fig. 5.20, draw a transversal t to the lines l and m such that one pair of corresponding angles is equal. You can measure the angles with a protractor. Are you finding it hard to draw a transversal such that the corresponding angles are equal?
Answer
Yes — and it is impossible, because the lines l and m in Fig. 5.20 are not parallel.
Try any transversal you like Corresponding angles always differ The difference stays the same for every transversal = the angle at which l and m would meet
Why it happens: Equal corresponding angles would force l and m to lean identically — that is, to be parallel. Since they are not parallel, no transversal can ever make a corresponding pair equal.
Did you know? This gives the rule the other way round too: when a pair of lines is not parallel, the corresponding angles formed by any transversal can never be equal. Equality of corresponding angles is therefore both necessary and sufficient for two lines to be parallel.