Bisecting an Angle
Geometry construction using a compass and straightedge

How to bisect an angle with compass and straightedge or ruler. To bisect an angle means that we divide the angle into two equal (congruent) parts without actually measuring the angle. This Euclidean construction works by creating two congruent triangles. See the proof below for more on this.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

This construction works by effectively building two congruent triangles. The image below is the final drawing above with the red lines added and points A,B,C labelled.

  Argument Reason
1 QA is congruent to QB They were both drawn with the same compass width
2 AC is congruent to BC They were both drawn with the same compass width
3 ∆QAC and ∆QBC are congruent Three sides congruent (sss). QC is common to both.
4 Angles AQC, BQC are congruent CPCTC. Corresponding parts of congruent triangles are congruent
5 The line QC bisects the angle PQR Angles AQC, BQC are adjacent and congruent

  - Q.E.D

Try it yourself

Click here for a printable worksheet containing three angle bisection problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Tangents

Ellipses

Polygons

Non-Euclidean constructions

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