Angle Bisector

Angle Bisector

What is it?

An angle bisector is the ray that splits an angle into two equal angles. To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector. In a triangle the three angle bisectors are concurrent. Their common point is the in-centre. The in-centre is always inside the triangle and is equidistant from the sides, so it is the centre of the incircle (the circle that touches all three sides).

In simple words

• An angle bisector is the ray that splits an angle into two equal angles.
• Use the same radius from the vertex to cut arm XP and arm XQ at P and Q.
• Drop a perpendicular from the in-centre to any side: that length is the in-radius.
• Equidistant from the sides means perpendicular distance, not distance to the vertices.

Important terms

equal angles: An angle bisector is the ray that splits an angle into two equal angles.

cut both arms: To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector.

equal arcs: To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector.

concurrent: In a triangle the three angle bisectors are concurrent.

in-centre: Their common point is the in-centre.

What you should learn

• Explain angle bisector in clear mathematical language.
• Use the main rule, formula, test, or construction correctly.
• Work through a simple example and show each important step.
• Identify common mistakes and write an exam-ready conclusion.

Examples

• Bisect a 70 deg angle into two 35 deg angles
• Do not change the first compass opening until both arms are cut
• The meeting of the second pair of arcs is not on the arms

DRAWING THE BISECTOR

Use the same radius from the vertex to cut arm XP and arm XQ at P and Q. Then, still using equal radii (they need not equal the first radius), swing an arc from P and from Q. Let the arcs meet at R. Ray XR bisects angle PXQ. The equal-arc step is what forces the two smaller angles to match.

IN-CENTRE AND INCIRCLE

Drop a perpendicular from the in-centre to any side: that length is the in-radius. The same length reaches the other two sides, which is why one circle can touch all three. Because each bisector stays inside its angle, the three bisectors always meet inside, even in an obtuse triangle. That is different from the orthocentre and the circumcentre, which can go outside.

Full school-level explanation

An angle bisector is the ray that splits an angle into two equal angles. To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector. In a triangle the three angle bisectors are concurrent. Their common point is the in-centre. The in-centre is always inside the triangle and is equidistant from the sides, so it is the centre of the incircle (the circle that touches all three sides).

Construction on one side of the page, the meeting point on the other:

WHAT — Core idea

An angle bisector is the ray that splits an angle into two equal angles. To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector. In a triangle the three angle bisectors are concurrent. Their common point is the in-centre. The in-centre is always inside the triangle and is equidistant from the sides, so it is the centre of the incircle (the circle that touches all three sides).

HOW — Rules, method, and examples

1. DRAWING THE BISECTOR: Use the same radius from the vertex to cut arm XP and arm XQ at P and Q. Then, still using equal radii (they need not equal the first radius), swing an arc from P and from Q. Let the arcs meet at R. Ray XR bisects angle PXQ. The equal-arc step is what forces the two smaller angles to match. 2. IN-CENTRE AND INCIRCLE: Drop a perpendicular from the in-centre to any side: that length is the in-radius. The same length reaches the other two sides, which is why one circle can touch all three. Because each bisector stays inside its angle, the three bisectors always meet inside, even in an obtuse triangle. That is different from the orthocentre and the circumcentre, which can go outside.

WHY — Meaning and connections

Equidistant from the sides means perpendicular distance, not distance to the vertices. Distance to the vertices is the circumcentre's job. Mixing in-centre with circumcentre is a frequent exam slip.

HOW — Answer in an exam

Bisector construction: one arc cuts both arms, then equal arcs from those points. Three bisectors meet at the in-centre, always inside, equidistant from the sides. That point is the centre of the incircle.

DRAWING THE BISECTOR

Use the same radius from the vertex to cut arm XP and arm XQ at P and Q. Then, still using equal radii (they need not equal the first radius), swing an arc from P and from Q. Let the arcs meet at R. Ray XR bisects angle PXQ. The equal-arc step is what forces the two smaller angles to match.

Remember

• same radius
• equal radii

Examples

• Bisect a 70 deg angle into two 35 deg angles
• Do not change the first compass opening until both arms are cut
• The meeting of the second pair of arcs is not on the arms

IN-CENTRE AND INCIRCLE

Drop a perpendicular from the in-centre to any side: that length is the in-radius. The same length reaches the other two sides, which is why one circle can touch all three. Because each bisector stays inside its angle, the three bisectors always meet inside, even in an obtuse triangle. That is different from the orthocentre and the circumcentre, which can go outside.

Remember

• in-radius
• touch
• always meet inside

Examples

• In-centre of an equilateral triangle is also the other centres
• In a scalene triangle the in-centre is still inside
• The incircle touches each side at one point, not at a vertex

Remember

• equal angles
• cut both arms
• equal arcs
• concurrent
• in-centre
• always inside
• equidistant from the sides
• perpendicular distance

How the ideas connect

Equidistant from the sides means perpendicular distance, not distance to the vertices. Distance to the vertices is the circumcentre's job. Mixing in-centre with circumcentre is a frequent exam slip.

Deeper understanding

Go beyond memorising Angle Bisector. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Equidistant from the sides means perpendicular distance, not distance to the vertices. Distance to the vertices is the circumcentre's job. Mixing in-centre with circumcentre is a frequent exam slip.

Common mistakes

Exam focus

WHAT

Key Points

HOW

Examples

Exam focus

Quick revision

• equal angles
• cut both arms
• equal arcs
• concurrent
• equal angles: An angle bisector is the ray that splits an angle into two equal angles.
• cut both arms: To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut poi

Bisector construction: one arc cuts both arms, then equal arcs from those points. Three bisectors meet at the in-centre, always inside, equidistant from the sides. That point is the centre of the incircle.

For long answers on Angle Bisector: define clearly → explain the main parts → give one example → state one exam caution.