To bisect an angle means to cut it into two equal angles with a straight line. With compasses and a ruler you can do it exactly, without measuring the angle at all.
This lesson follows constructing a perpendicular bisector in constructions and loci. It is also the answer to “find the points equidistant from two lines”.
Why does the construction work?
The first arc, centred on the vertex B, marks two points P and Q on the arms that are the same distance from B. The second pair of arcs, centred on P and Q with equal radius, meet at R. So BP = BQ and RP = RQ, and BR is shared.
The two triangles are congruent, which means the angles at B are equal.
A useful consequence: every point on the bisector is the same distance from both arms.
How do you construct it, step by step?
- Draw the angle with vertex B and arms BA and BC.
- Place the compass point on B and draw an arc that crosses both arms. Label the crossings P and Q.
- Place the point on P and draw an arc inside the angle.
- Keep the same radius, place the point on Q and draw an arc that crosses the previous one at R.
- Join B to R with a ruler and extend it if needed. The line BR is the angle bisector.
Worked example
Construct the bisector of angle ABC = 70°, then check it.
Step 1, draw the angle: with a protractor, draw BA and BC so that the angle is 70°.
Step 2, mark P and Q: with the compass point on B and radius 4 cm, draw an arc across both arms.
Step 3, mark R: keep the compasses at a radius larger than half of PQ. Draw arcs from P and Q that cross at R.
Step 4, draw the line BR.
Step 5, check by measuring: half of 70° is 35°, so angle ABR and angle RBC should each measure 35°, within about 1°.
Step 6, check by distance: any point on BR should be the same perpendicular distance from BA and BC. Test one point near the end.
The mistake to watch for
A common slip is to use a different radius for the two second arcs, or to move the compass point off P or Q while drawing.
Mistaken result: the bisector measures 32° and 38° instead of 35° and 35°.
The radius changed between the arcs, so R is nearer to one arm.
The correction is to hold the compasses steady after the first arc and redraw both second arcs with the same radius. Then measure both halves, not just one.
Check yourself
Try each question, then open the answer.
1. Angle PQR = 50°. After bisecting it, what is each of the two angles?
Show answer
50° ÷ 2 = 25° each.
2. Point X is on the bisector of angle ABC and is 6 cm from arm BA, measured at right angles. How far is X from arm BC?
Show answer
Every point on an angle bisector is equally far from both arms, so X is 6 cm from BC.
3. An angle of 120° is bisected. A point on the bisector is 5 cm from one arm. What angle does the bisector make with each arm, and how far is the point from the other arm?
Show answer
Each half is 120° ÷ 2 = 60°. The point is on the bisector, so it is also 5 cm from the other arm.
Where this leads next
Both constructions become tools once you turn a distance rule into a shape, as in describing a locus using distance conditions. The non-calculator working trainer helps with the halving and checking arithmetic.
Some students draw neat arcs but are unsure when to use the angle bisector instead of the perpendicular bisector. Our teachers work through that choice in online one-to-one Mathematics tuition.