Definition: An angle bisector is a ray from the vertex of an angle that splits it into two congruent angles, each half of the original measure.
What does a bisector do?
If ray BD bisects angle ABC, then angle ABD and angle DBC have the same measure, and each equals half of angle ABC. Every point on the bisector is the same distance from the two sides of the angle.
How do you use the definition?
Set the two smaller angles equal and solve. Or double one half to find the whole angle.
Example. Ray BD bisects angle ABC. Angle ABD = (6x – 5) degrees and angle DBC = (4x + 15) degrees. Setting them equal: 6x – 5 = 4x + 15, so 2x = 20 and x = 10. Each half is 55 degrees, so angle ABC = 110 degrees.
What mistakes do students make?
They mix up bisecting an angle with bisecting a segment. A segment is divided by a midpoint into equal lengths, while an angle is divided by a ray into equal measures. Another mistake is to use the unknown value as the angle: x = 10 is not the angle measure. You can construct a bisector with a compass: arcs from the two sides meet at a point on the bisector.
Where to learn it
- Points, Segments and Angles: Core Definitions – lesson
- Compass and Straightedge Constructions – lesson