Comparing angles

Vidya opens the cover of her book, a little at first, then wider and wider. The more she turns the cover, the bigger the angle. We describe a line's size by its length, but an angle's size is the amount of turn about the vertex needed to move one arm onto the other. You could not tell someone a clock hand 'moved 2 metres'; you say how much it turned. Animals opening their jaws work the same way: the wider the jaws turn apart, the larger the angle.

The length of the arms does not matter. Draw the same angle with longer arms, or shorter ones: the turn between them is unchanged, so the angle is unchanged.

One way to compare two angles is superimposition: put one angle on top of the other so that the vertices sit together and one arm of each lies along the other. Then look at the second arms:

• In your book, ∠PQR is placed on ∠ABC, with Q on B and arm QP along BA. Arm QR falls outside BC, so ∠PQR is larger.
• If the second arms also lie exactly on top of each other, the angles are equal, as ∠AOB and ∠XOY are.

Two cranes argue about who opens its beak wider. To compare without tracing, use a transparent circle: put its centre on the vertex of the first angle and mark where the arms cross the edge (A and B). Then put the circle on the second angle, centre on its vertex and one arm through A, and see where the other arm crosses: before B, at B, or beyond it.

In another activity, pairs of rotating arms are made from two straws and a paper clip, and a slit in cardboard is cut in the shape of one angle. If the slit's angle is bigger than the arms' angle, the arms do not go through; if it is smaller, they do not go through either. Only arms at exactly the slit's angle pass through, however long or short they are (as long as they are shorter than the slit).