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Angle Bisector

An angle bisector is a line or ray that divides an angle into two congruent angles .

Math diagram

In the figure, the ray $\vec{K M}$ bisects the angle $∠ J K L$ .

The angles $∠ J K M$ and $∠ L K M$ are congruent.

So, $m ∠ J K M = m ∠ L K M$ .

Note that any point on the angle bisector is equidistant from the two sides of the angle. Some textbooks call this Angle Bisector Theorem , but this name is usually used for another theorem about angle bisectors in a triangle.

Example :

Math diagram

In the figure, $\vec{B D}$ is an angle bisector. Find the measure of $∠ A B C$ .

Since $\vec{B D}$ is an angle bisector,

$m ∠ C B D = m ∠ A B D$

Therefore,

$m ∠ C B D = 65 °$

By the Angle Addition Postulate,

$m ∠ A B C = m ∠ A B D + m ∠ C B D$ .

Substitute.

$\begin{array}{l} m ∠ A B C = 65 ° + 65 ° \\ = 130 ° \end{array}$