Worked Example 1
Construct an angle equal to a given angle ABC
| Step | What you do |
|---|---|
| 1 | With B as centre and any radius, draw an arc cutting BA at P and BC at Q. |
| 2 | Draw a ray YZ. With Y as centre and the same radius, draw an arc cutting YZ at M. |
| 3 | Set the compass to PQ. With M as centre and this width, cut the arc at N. |
| 4 | Join YN. Then angle NYZ = angle ABC. |
Reason to write downIn triangles BPQ and YMN: BP = YM (same radius), BQ = YN (same radius), PQ = MN (copied). So the triangles are congruent by SSS, and therefore angle B = angle Y.
Worked Example 2
Divide a given angle into four equal parts
Think firstYou cannot cut into four in one go. But four is two twos. Bisect once to get two halves, then bisect each half. Two rounds of bisecting, three new rays in total.
StepsBisect angle AOB with ray OX. Now bisect angle AOX with ray OY, and bisect angle XOB with ray OZ. The four angles AOY, YOX, XOZ and ZOB are all equal.
The follow up questionCan you divide an angle into eight equal parts? Yes, three rounds of bisecting. Into sixteen? Yes, four rounds. Into three? No, and no amount of bisecting will ever get you there, because halving only ever produces halves, quarters, eighths and so on.