What Is the Triangle Angle Bisector Theorem?


Quick Answer

The angle bisector theorem states that a line bisecting an angle in a triangle divides the side opposite the angle into two line segments that have lengths proportional to the lengths of the other sides. An angle bisector is a line that divides an angle into two equal angles; it is often depicted as a ray emanating from an angle's vertex.

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Full Answer

To give the result of the angle bisector theorem more precisely, it is necessary to label the parts of a triangle. If one calls the points of a triangle x, y and z, one can say that a line bisects the angle with its vertex at point x and that this line goes through the opposite side of the triangle at point w. The line segments on the triangle are named according to their endpoints, so the side of the triangle between points x and y is termed segment XY, for example.

According to the angle bisector theorem, the length of segment YW divided by the length of segment WZ is equal to the length of segment YX divided by the length of segment ZX. The proof of this theorem relies on the law of sines, which comes from trigonometry.

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