Q1.
Take a square sheet of paper, fold it in the middle and unfold it. Fold the edges towards the centre line and unfold them. Fold the top right and bottom left corners onto the creased line to create triangles. Refer to Fig. 5.8. The triangles should not cross the crease lines. Are a, b and c parallel to p, q and r respectively? Why or why not?
Answer
Yes. a ∥ p, b ∥ q and c ∥ r.
| Pair | What the two sides are | Why they are parallel |
|---|---|---|
| a and p | The vertical sides of the two triangles, lying along crease lines | All the creases run parallel to the vertical edges of the square |
| b and q | The horizontal sides of the two triangles | Both are perpendicular to those vertical creases, so they point the same way |
| c and r | The two fold lines (the slanting sides) | Each is the diagonal of an equal square, so each makes the same 45° with the edges |
Why it happens: The two corner folds are made in exactly the same way at opposite corners. So the whole picture looks unchanged if you turn the sheet through half a turn (180°) about its centre — the top-right triangle lands exactly on the bottom-left one. A half-turn always sends a line to a line pointing in the same direction, that is, to a parallel line. That single fact explains all three pairs at once.
Check it yourself: Trace triangle abc on tracing paper, put a pin through the centre of the square, and rotate the tracing by 180°. It falls exactly on triangle pqr.