Angle as an amount of turning
What an angle is, what a degree measures, and why the vertex and the arm lengths do not matter.
Point, line, segment, midpoint: all of them so far are about where things are. This lesson is about something different: how far one direction has turned from another.
Here is a vertex, O, with two arms reaching out to A and B, and a reading of the angle between them underneath. Drag either arm and watch what moves the reading and what doesn’t.
A vertex, O, with two arms running out to A and B. All three can be dragged. This needs JavaScript switched on.
The reading passes through 90 on its way from small to large, and again through 180 when B swings all the way round to point opposite A. Try to make the two arms point in exactly the same direction and the reading should hit 0.
Where two arms meet is the vertex of the angle: O, in the widget above. The two straight paths running out from it are its arms: OA and OB. The angle itself is not a length and not a point. It is the amount of turning it takes to swing one arm round onto the other.
That amount is measured in degrees, written 90° rather than “90 degrees” from here on. A full turn, all the way round back to where you started, is 360 degrees. Half of that, a straight line continuing through the vertex with no turn at all sideways, is a straight angle, 180°. A quarter turn, exactly square, is a right angle, 90°, and it gets its own small square mark instead of a curved one, because it comes up often enough to deserve one.
This is why dragging an arm further out changed nothing: an angle is fixed by two directions, not by four points. OA and OB could be drawn twice as long or half as long and the angle between them would not move, because turning is a property of direction, not of distance — the widget was showing you the one thing that does not depend on how far out you put the point. The vertex doesn’t matter to the reading either, for the same reason: slide the whole angle sideways, vertex and both arms together, and the turn between the two directions is untouched.
A right angle and a straight angle are not measured so much as reasoned to. Half of a full turn is a straight angle by definition of what “half” means, and a quarter turn is a right angle the same way. A reading taken off a protractor, or off the widget above, is a measurement, and a measurement is only ever approximate. Knowing that a straight line is 180° exactly is not a measurement at all: it is what “straight” and “half a turn” already agreed to mean.
A reading like “about 47 degrees” is honest about being approximate. “180 degrees, because it is a straight line” is not a reading at all. Later lessons build almost everything on the second kind, not the first.
Two arms pointing in exactly opposite directions. What is the angle between them?
Opposite directions is a half turn, no matter how long either arm is drawn:
One arm pointing east, the other pointing straight up.
A quarter turn from one to the other, so a right angle:
An arm dragged from 3 squares out to 9 squares out, direction unchanged. What happens to the reading?
Nothing. The angle is about the two directions, and neither direction moved. Only the arms got longer.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.