Angle Bisectors

An angle bisector is a line or ray that divides an angle into two equal parts.

Each angle formed by the bisector is $$\frac{\theta}{2}$$ .

$$m\text{angle ABD} = m\text{angle DBC}$$ .

The angle bisector divides the opposite side into segments that are proportional to the lengths of the other two sides.

The ratio $$\frac{AD}{DC} = \frac{AB}{AC} = \frac{4}{6} = \frac{2}{3}$$ .

To find the measure of an angle when given its bisector, you can use the property that the bisector divides the angle into two equal parts.

$$m\text{angle PQS} = \frac{m\text{angle PQR}}{2} = \frac{104\text{°}}{2} = 52\text{°}$$ .

The angle bisector theorem states that the angle bisector of a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides.

$$m\text{angle DBC} = 9\text{°}$$ .

Using the angle bisector theorem, $$\frac{AB}{AC} = \frac{BD}{DC}$$ , so $$\frac{5}{10} = \frac{3}{DC}$$ , thus $$DC = 6$$ .