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Angle bisectors

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Angle Bisectors

An angle bisector is a ray that divides an angle into two equal smaller angles, and every point on it is equidistant from the angle's two sides. Learn how to construct one with compass and straightedge, and how the angle bisector theorem splits the opposite side of a triangle in proportion to the other two sides.

What an angle bisector is

An angle bisector is a ray that divides an angle into two equal smaller angles. If a bisector cuts a 70° angle, it creates two 35° angles. Every point on the bisector is the same distance from the two sides of the angle.

An angle bisector splits an angle into two equal angles Two rays form an angle at a vertex. A third ray, the angle bisector, runs between them and divides the angle into two equal parts, shown by equal angle marks on each side. bisector A B
An angle bisector splits the angle into two equal parts, marked as equal.

How to construct one

With a compass and straightedge: place the compass at the vertex and draw an arc crossing both sides; from each crossing point draw two more arcs that meet inside the angle; the ray from the vertex through that meeting point is the bisector. This construction is a close cousin of the perpendicular bisector of a segment.

The angle bisector theorem

In a triangle, an angle bisector divides the opposite side into two segments that are proportional to the other two sides. If the bisector from angle A meets side BC, then BD/DC = AB/AC. This is a standard result once you are comfortable with pairs of lines and angles.

Why it matters

Angle bisectors locate the incenter of a triangle (where all three meet) and appear throughout constructions and proofs, building on basic angle relationships.

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