Copying an angle

Some figures are built from a single UNIT repeating itself, in different orientations. To construct one, we must make exact copies of that unit.

For copies to be exact, all the units must have the same arm lengths and the same ANGLE between the arms. Equal arm lengths are easy with a compass. But how do we guarantee an equal angle, without a protractor?

We need a way to copy an angle. Here is one, and every step uses only the ruler and compass.

Step 1: you have an angle at A, and you want a copy at a point X on a line. Step 2: draw an arc from A cutting both arms, at B and C. Notice what you now have: AB and AC are both radii of that arc, so they are equal, and ∆ABC is an isosceles triangle.

Step 3: draw an arc of the SAME radius from X, cutting the line at Z. Since the radius is unchanged, XZ = AB = AC.

Step 4: measure the distance BC with the compass, and transfer it onto the new arc, stepping from Z to get a point Y with YZ = BC.

Step 5: now compare ∆ABC and ∆XYZ. All three pairs of sides are equal — AB = XZ, AC = XY, and BC = ZY — so by the SSS congruence condition the triangles are congruent, and therefore ∠A = ∠X. The angle has been copied exactly, with no measurement in degrees or centimetres anywhere.