Brief Analyses on Angle Bisector Theorem

According to the angle bisector theorem, a triangle’s opposite side is divided into two segments that are proportional to the triangle’s other two sides. The ray that divides a given angle into two equal halves is known as an angle bisector of the given angle. In this post, we’ll study more about the angle bisector theorem.

The angle bisector of a triangle bisects the opposite side in such a way that the ratio of the two-line segments is proportionate to the ratio of the other two sides, according to the Angle Bisector theorem. As a result, the opposite side’s relative lengths (split by angle bisector) are equal to the lengths of the triangle’s other two sides. The angle bisector theorem can be applied to any triangle.

What is the Angle Bisector Theorem?

According to the triangle angle bisector theorem, the angle bisector of every angle splits the opposite side in the ratio of the sides containing the angle. Consider the figure below :

PS is the bisector of ∠P in this case. PQ/PR = QS/RS, or a/b = x/y, according to the angle bisector theorem.

A line or a ray that divides an angle in a triangle into two equal measures is called an angle bisector. The angle bisector property of a triangle states that every point on the bisector of an angle is equidistant from the sides of the angle and that the angle bisector splits the opposing side of a triangle in the ratio of the neighbouring sides.

Angle Bisector Theorem Proof

According to the basic proportionality theorem, if a line is drawn parallel to one of a triangle’s sides and intersects the other two sides in distinct spots, the other two sides are divided in the same ratio.

DA and CE are parallel in ∆CBE.

BA/AE = BD/DC …. ..(1)

It’s now up to us to show that AE = AC.

Let’s draw the angles in the diagram above.

We have DA because it is parallel to CE.

∠DAB = ∠CEA (corresponding angles) —- (2)

∠DAC = ∠ACE (alternate interior angles) —- (3)

We have DAB = DAC since AD is the bisector of BAC (4).

We can say that ∠CEA = ∠ACE because of (2), (3), and (4). It gives ∆ACE the shape of an isosceles triangle. We have AC = AE because sides opposing equal angles are equal.

Substitute AC for AE in equation (1).

BD/DC = BA/AC

Hence proved.

Converse of Angle Bisector Theorem

If the sides of a triangle satisfy the criteria “If a line drawn from a vertex of a triangle divides the opposite side into two pieces such that they are proportional to the other two sides of the triangle,” it implies that the point on the opposite side of that angle falls on its angle bisector. Because sides are in proportion, we can conclude that the line/ray/segment is the angle bisector of the relevant angle. In geometry, this is known as the converse of the angle bisector theorem. If a side of a triangle is sliced into two parts that are proportional to the sides of the triangle that each part intersects, the angle across from the segmented side is bisected by the line that cuts the segmented side into two parts, according to the converse of the angle bisector theorem.

Angle Bisector Theorem Formula

According to the triangle angle bisector theorem, the angle bisector of every angle splits the opposite side in the ratio of the sides containing the angle. The triangle angle bisector theorem formula can be written as, 

BD\DC=AB\AC

Conclusion

The angle bisector theorem is useful in percentage problems and proofs because it may be used to establish that two line segments have a certain proportion.

• The search for a missing side length.
• In search of an angle bisector.
• Demonstrating the similarity of polygons
• Working with the geometric mean and the property of means-extremes.

Working with a transversal to cut the parallel lines.

The angle bisector theorem and its converse are used to find a missing side length and to check for an angle bisector. The angle bisector theorem may be useful in establishing a proportion or segment length that can be employed later in the problem or proof in the other three cases described above.