Constructions
57. The Angle Splitter · cutting any angle perfectly in half
OE always splits ∠AOB into two equal angles, no matter how you drag the arms.
To bisect an angle with compass and straightedge: from O, draw an arc cutting both arms at equal distances. From those two intersection points, draw equal arcs that meet at E. The line OE bisects ∠AOB into two equal halves.
What this lesson covers
Try to break it
What if we change the angle itself? Drag O, A, or B to reshape the angle, and see how the bisector OE always stays perfectly balanced!
How you build it
Construct the bisector of an angle.
- Mark point O — the vertex of the angle.
- Draw a ray from O — the first arm.
- Draw a second ray from O — the other arm. This is the angle to bisect.
- With centre O, draw an arc that cuts both arms.
- Mark point C where the arc cuts the first arm.
- Mark point D where the arc cuts the other arm.
- With the same opening and centre C, draw an arc inside the angle.
- With the same opening and centre D, draw an arc crossing the one from C.
- Mark point E where the two arcs cross.
- Draw the ray OE — it bisects the angle into two equal halves.
The proof, step by step
Prove that ray OE divides ∠AOB into two equal angles.
- In ΔOCE and ΔODE, OC = OD (radii of the same arc drawn from O).
- CE = DE (arcs drawn with equal radii from C and D).
- OE = OE (common side).
- Therefore, ΔOCE ≅ ΔODE by SSS congruence rule.
- Hence, ∠COE = ∠DOE (c.p.c.t.).
- So, OE bisects ∠AOB.
Worked example
In the construction of the bisector of ∠AOB, arcs are drawn from C and D with equal radii. Why must this radius be more than half of CD?
If the radius is less than or equal to half of CD, the arcs from C and D will not intersect. We need point E to draw the bisector OE.
- To make the drawing look neater
- So that the two arcs intersect each other — correct
- To ensure the compass doesn't slip
- It does not matter what the radius is