How to Split an Angle Exactly in Half Without a Protractor

Yes — a pair of compasses does it exactly, and it works even for an angle whose size you never find out. Two arcs and a straight line are enough.

That is what makes constructions different from measuring: a protractor is accurate to about half a degree, while a construction is exact by its geometry. This page covers everything in the CBSE Class 7 Mathematics chapter's first half: perpendicular bisectors, angle bisectors, standard angles, parallel lines, and designs built from arcs.

The perpendicular bisector of a segment cuts it in half at right angles, and two pairs of arcs produce it.

To bisect AB:

1. Open the compasses to more than half of AB.
2. With the point at A, draw arcs above and below the line.
3. Keeping the same opening, repeat from B.
4. The arcs cross at two points; join them with a ruler.

That line cuts AB at its midpoint and meets it at $90°$. So for AB = 8 cm, each half is 4 cm without measuring either half.

The reason it works: every point on the new line is the same distance from A as from B, and the only such points lie on the perpendicular bisector.

Step 1 carries the boundary case. If the opening is less than half of AB, the arcs from A and B never reach each other and there is nothing to join. Arcs that refuse to cross mean the compasses are too tight, not that the drawing is wrong.

To drop a perpendicular at a given point P on a line, first mark two points equally distant from P on either side, then bisect the segment they form — the bisector passes through P at $90°$.

To bisect $\angle ABC$: place the compass point at B and draw an arc cutting both arms, at P and Q. Then, from P and Q in turn with the same opening, draw two arcs that cross at R. The ray BR is the bisector.

The standard angles come from two starting points and repeated bisection.

**$60°$: draw a base line, mark B, and with any opening draw an arc from B cutting the line at P. With the same opening** and the point at P, draw an arc crossing the first at Q. Then $\angle QBP=60°$.

**$90°$**: construct a perpendicular to the line at B.

Everything else is halving:

$$\frac{90}{2}=45°,\frac{60}{2}=30°,\frac{30}{2}=15°$$

So $45°$ is the bisector of a right angle, $30°$ the bisector of $60°$, and $15°$ the bisector of that $30°$.

A folded sheet of paper does the same job for a right angle — fold once, then fold the crease onto itself — which is the everyday version of bisecting.

The limit is worth knowing: repeated halving of $60°$ and $90°$ gives $45,30,22.5,15°$ and so on, but never $20°$. Some angles simply cannot be built with ruler and compasses alone, and a protractor is the honest tool for those.

By copying an angle. Parallel lines make equal corresponding angles with any line crossing them, so reproducing that angle at a new point produces a parallel.

To draw a line through P parallel to line $l$:

1. Draw any line through P that crosses $l$, meeting it at Q. This is the transversal.
2. At Q, draw an arc cutting both $l$ and the transversal.
3. With the same opening, draw a matching arc at P.
4. Set the compasses to the width of the arc's cut at Q, transfer that width to the arc at P, and mark the point.
5. Join P to that point — the line is parallel to $l$.

What you have done is copy the angle at Q up to P. Equal corresponding angles guarantee the two lines never meet.

Alternate angles work just as well, provided you copy the angle on the opposite side of the transversal. Copying the correct angle onto the wrong side produces a line that slopes the other way and crosses $l$ — the usual error here.

Ruled lines in a notebook are the everyday case: every line makes the same angle with the margin, which is exactly why they stay the same distance apart all the way down the page.

Because the compass radius steps round its own circle exactly six times, and each step makes an equilateral triangle.

Draw a circle of radius 4 cm. Without changing the opening, place the compass point anywhere on the circle and mark an arc; move to that mark and repeat. Six marks bring you back to the start, and joining them gives a regular hexagon of side 4 cm.

The angle at the centre for each step is

$$\frac{360}{6}=60°$$

which is the angle of an equilateral triangle — so the radius fitting six times is no coincidence.

From the same six points:

- Joining every second point gives a triangle, and two such overlapping triangles make a six-pointed star.
- Drawing an arc inside the circle from each of the six points produces the trefoil and flower patterns seen in window grills and jaali screens.
- Two arcs of equal radius drawn from the two ends of a segment meet in a point above it, giving a pointed arch of the kind used in old gateways.

Keep the compass opening unchanged throughout a design. Reopening the compasses even slightly makes the six steps miss the starting point, and the pattern will not close.