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.
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.
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.
|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|