Is it possible for all the eight angles to have different measurements? Why, why not?
Answer
No, it is not possible. The eight angles can show at most four different sizes.
At line l: ∠1 = ∠3 and ∠2 = ∠4 (vertically opposite) At line m: ∠5 = ∠7 and ∠6 = ∠8 (vertically opposite) 8 angles → grouped into 4 equal pairs So at most 4 different measures
Transversal t cuts lines l and m, making eight angles. ∠1, ∠2, ∠3, ∠4 at the first crossing; ∠5, ∠6, ∠7, ∠8 at the second.
Why it happens: At each crossing the four angles are not independent — the facing ones must be equal and neighbours must add to 180°. So each crossing offers only two different sizes, and two crossings give at most 2 + 2 = 4.
Did you know? If lines l and m happen to be parallel, even those four sizes collapse to two: one acute and one obtuse, adding to 180°.
Q2.
What about five different angles — 6, 5, 4, 3 and 2?
Answer
Five is not possible either. Nor is three or six.
Possible numbers of different measures = 1, 2 or 4 4 → the general case (l and m not parallel) 2 → l ∥ m, transversal slanting 1 → t perpendicular to both l and m (all eight angles 90°)
To get five different values you would need at least one crossing to show three different angles — impossible, because ∠2 = ∠4 and ∠1 = ∠3 always.
Why it happens: Angles come in forced pairs. Once you fix one angle at a crossing, the other three are decided — one equals it, and the other two are its supplement. So the count of different sizes can only be 1, 2 or 4.
Q3.
In Fig. 5.14, since ∠1 and ∠3 are vertically opposite angles, they are equal. Are there other pairs of vertically opposite angles?
Answer
Yes — four pairs in all, two at each crossing.
Crossing
Vertically opposite pairs
Relation
t with line l
∠1 and ∠3
equal
t with line l
∠2 and ∠4
equal
t with line m
∠5 and ∠7
equal
t with line m
∠6 and ∠8
equal
Why it happens: Each crossing of two lines is a complete little picture of its own, and every such crossing has exactly two pairs of facing angles. Two crossings therefore give 2 × 2 = 4 pairs.
Tip: Do not confuse these with corresponding angles — ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 — which sit in the same position at the two different crossings.