Geometric Construction
With only a compass and a straightedge you can copy, bisect and build exactly — and the angle facts tell you when a figure is even possible.
Set a compass to some width and do not touch the screw again.
Every arc you now draw is made of points at exactly that distance from where the point sat.
Put the point at one end of a line and sweep. Put it at the other end, same width, and sweep again.
Where the arcs cross is a point that is the same distance from both ends — and it was found by the tool, not measured by you.
Join it up and every side is equal, because every side was that one compass width.
That is why constructions use a compass and a straightedge and nothing else. A ruler measures and can be misread; a compass simply keeps a distance, exactly, for as long as you leave it alone.
The angle facts you build on
Angles on a straight line sum to 180°. Angles around a point sum to 360°.
Vertical angles are equal. When two lines cross, the opposite angles match.
With parallel lines and a transversal: corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°.
A triangle's angles sum to 180°; a quadrilateral's to 360°.
Angle chasing
Most construction and geometry problems are solved by filling in one angle at a time, each justified by a fact above.
Work from what you know outward. Mark every angle you find directly on the figure — the next step usually depends on it.
Write the reason beside each angle
Do not just write 70°. Write "70° — vertical angles". The reason is what tells you which angle you can find next, and it is what earns credit on a test.
How many triangles fit the clues?
Given three sides: exactly one triangle, if the triangle inequality holds. Sides 3, 4, 9 make nothing, because 3 + 4 < 9 — the short sides cannot reach.
Given two sides and the angle between them: exactly one triangle.
Given three angles: infinitely many triangles, all similar but different sizes.
Given two sides and an angle not between them: sometimes two different triangles fit.
Triangle inequality
Three lengths form a triangle exactly when the two shortest add to more than the longest. Check only that one comparison — it is enough.
The four core constructions
Copy a segment: set the compass to its length, then strike that length off on a new ray.
Bisect a segment: from each endpoint, draw arcs of the same radius (more than half the length). They cross above and below; the line joining those crossings cuts the segment in half — and at a right angle.
Bisect an angle: from the vertex, draw an arc crossing both sides. From those two crossing points, draw equal arcs that meet. Join the vertex to that meeting point.
Perpendicular through a point: a bisected segment gives you a right angle for free.
Why the constructions work
Every one of them relies on the same idea: a compass draws all the points at an equal distance from its centre.
When two equal arcs cross, that crossing point is the same distance from both starting points — which is exactly what "halfway" means. The construction is a proof, not a trick.
Worked examples
Angle chasing
Two lines cross. One angle measures 52°. Find the other three.
- The vertical angle is also 52°. Vertical angles are equal.
- A neighbouring angle: 180° - 52° = 128° They sit on a straight line.
- Its vertical angle is 128° too. Vertical angles again.
- Check: 52 + 52 + 128 + 128 = 360 ✓ Angles around a point sum to 360°.
52°, 128°, 128°.
Can these be a triangle?
Can segments of length 5, 7 and 13 form a triangle?
- Identify the longest side: 13. Only one comparison matters.
- Add the other two: 5 + 7 = 12 The two shorter sides together.
- 12 < 13 The short sides cannot reach across the long one.
No — the triangle inequality fails.
Parallel lines
Two parallel lines are cut by a transversal. One angle is 73°. Find the co-interior angle on the same side.
- Co-interior angles sit between the parallels, on the same side of the transversal. Identify the right pair before computing.
- They are supplementary: they sum to 180°. This is the co-interior angle rule.
- 180° - 73° = 107° Subtract.
107°
Words to know
- Compass / straightedge
- Tools for drawing exact circles and lines — no measuring allowed.
- Bisect
- Cut exactly in half.
- Complementary / supplementary
- Two angles summing to 90° / to 180°.
- Vertical angles
- The opposite pair formed by two crossing lines — always equal.
- Triangle inequality
- Any two sides must sum to more than the third.
Mistakes to watch for
Changing the compass width in the middle of a construction. The equal radii are the entire point — they are what guarantees the sides come out equal. Once the screw moves, the proof is gone.
Checking all three comparisons in the triangle inequality. Only the longest side matters: if it is shorter than the other two added together, the triangle closes. The other two checks pass automatically.
Assuming two angles are equal because they look equal. A drawing is evidence of nothing; name the rule that makes them equal — vertical, corresponding, or base angles of an isosceles triangle.
Rubbing out the construction arcs. They are the working, and they show why the figure is right. Marks for a construction are usually for the arcs, not the finished shape.
The Greeks asked whether you could trisect any angle, double a cube, or square a circle with compass and straightedge alone. People tried for two thousand years.
In the 1800s all three were proved impossible — not hard, impossible — using algebra that had not existed when the questions were asked.
Gauss, aged 19, did construct a regular 17-sided polygon, which nobody had managed. He asked for it on his gravestone.