How To Bisect An Angle With Compass And Straightedge Or Ruler

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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.

Other constructions pages on this site

• List of printable constructions worksheets

Lines

• Introduction to constructions
• Copy a line segment
• Sum of n line segments
• Difference of two line segments
• Perpendicular bisector of a line segment
• Perpendicular at a point on a line
• Perpendicular from a line through a point
• Perpendicular from endpoint of a ray
• Divide a segment into n equal parts
• Parallel line through a point (angle copy)
• Parallel line through a point (rhombus)
• Parallel line through a point (translation)

Angles

• Bisecting an angle
• Copy an angle
• Construct a 30° angle
• Construct a 45° angle
• Construct a 60° angle
• Construct a 90° angle (right angle)
• Sum of n angles
• Difference of two angles
• Supplementary angle
• Complementary angle
• Constructing 75° 105° 120° 135° 150° angles and more

Triangles

• Copy a triangle
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• 30-60-90 triangle, given the hypotenuse
• Triangle, given 3 sides (sss)
• Triangle, given one side and adjacent angles (asa)
• Triangle, given two angles and non-included side (aas)
• Triangle, given two sides and included angle (sas)
• Triangle medians
• Triangle midsegment
• Triangle altitude
• Triangle altitude (outside case)

Right triangles

• Right Triangle, given one leg and hypotenuse (HL)
• Right Triangle, given both legs (LL)
• Right Triangle, given hypotenuse and one angle (HA)
• Right Triangle, given one leg and one angle (LA)

Triangle Centers

• Triangle incenter
• Triangle circumcenter
• Triangle orthocenter
• Triangle centroid

Circles, Arcs and Ellipses

• Finding the center of a circle
• Circle given 3 points
• Tangent at a point on the circle
• Tangents through an external point
• Tangents to two circles (external)
• Tangents to two circles (internal)
• Incircle of a triangle
• Focus points of a given ellipse
• Circumcircle of a triangle

Polygons

• Square given one side
• Square inscribed in a circle
• Hexagon given one side
• Hexagon inscribed in a given circle
• Pentagon inscribed in a given circle

Non-Euclidean constructions

• Construct an ellipse with string and pins
• Find the center of a circle with any right-angled object

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