Understanding the Angle Bisector Theorem | Proportional Relationships in Triangles

Theorem 6.3: Angle Bisector Theorem

The Angle Bisector Theorem states that if a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the lengths of the two adjacent sides

The Angle Bisector Theorem states that if a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the lengths of the two adjacent sides.

To illustrate this theorem, let’s consider a triangle ABC, where AD is the angle bisector of angle A, with point D lying on side BC. According to the Angle Bisector Theorem, we have the following proportion:

BD/DC = AB/AC

This means that if we divide side BC into two segments, BD and DC, the ratio of their lengths is equal to the ratio of the lengths of the sides adjacent to angle A.

Furthermore, we can rearrange the terms of this equation to obtain other proportional relationships. For example:

AB/BD = AC/DC

In this case, by swapping the positions of the sides, we can determine the proportionality between the lengths of the side adjacent to angle A with respect to the lengths of the segments on side BC.

The Angle Bisector Theorem is a useful tool in geometry, as it allows us to determine unknown lengths or segment ratios using known values. It can be applied to solve various problems involving angles and triangle geometry.

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