Understanding angles (age 12+)
Use ruler and compasses to perform standard constructions: perpendicular bisector of a line segment, perpendicular to a line from or at a given point, and bisecting an angle
What a learner can do afterwards
- Construct the perpendicular bisector of a line segment using compasses and a straight edge
- Construct an angle bisector and verify by measuring both halves
- Construct a perpendicular from a point to a line using compasses only
The lesson
A construction is a drawing made with only two tools: a ruler as a straight edge, and a pair of compasses. You never use a protractor. Every construction works because a compass draws every point that sits the same distance from one center point.
To bisect segment AB exactly in half: open your compasses wider than half of AB. Put the point on A and draw an arc above and below the line. Keep the same width, put the point on B, and draw two more arcs that cross the first ones. Draw a straight line through the two crossing points. That line is the perpendicular bisector: it cuts AB in half at a right angle.
To drop a perpendicular from point P to a line: put the compass point on P and draw an arc that crosses the line at two spots. From each of those spots, draw an arc below the line, keeping the same width. Where those two arcs cross, draw a line back to P. That line meets the original line at exactly 90 degrees.
To bisect an angle at point O: put the compass point on O and draw an arc that crosses both arms. Keeping the same width, put the point on each crossing spot and draw two more arcs that meet inside the angle. Draw a line from O through that meeting point. It splits the angle into two equal halves.
Keep your compass width exactly the same for both arcs in one step. If you widen or narrow it partway through, the arcs will not cross where they should, and the construction comes out wrong.
Same compass width, two crossing arcs, then a straight line through the crossing point: that is how you build an exact half or a true right angle without ever using a protractor.
Watch it
Where it sits
Learn first
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.