Angle Bisector Theorem

Introduction:

The angle bisector theorem can be defined as the relative lengths of the two segments that a line bisects the opposite angle divides a triangle’s side into. It is equal to their relative lengths to the other two sides of the triangle’s relative lengths. A ray, segment, or line that divides a given angle into two equal angles is known as an angle bisector. The term bisector or bisection refers to the division of a single object into two equal halves. In geometry, a line or ray is used to split a triangle and an angle, and this is known as an angle bisector. In geometry, an angle bisector is a ray, line, or segment that divides an angle into two equal pieces. A 60-degree angle, for example, will be divided into two 30-degree angles by an angle bisector. This divides 1 angle into 2 smaller congruent angles, in other words. Because an angle bisector divides the side opposite that angle into two segments that are proportional to the triangle’s other two sides, the Angle Bisector Theorem can be used to calculate unknown lengths of triangle sides. The internal bisector of an angle of triangle divides the opposing side internally in the ratio of the angle’s corresponding sides.

Properties of Angle Bisector:

We should now understand what angle bisector means in geometry. Let’s look at some of the qualities of the angle bisector, which are stated below:

1. An angle bisector is a tool that divides a given angle into two equal halves.
2. Any point on an angle’s bisector is equidistant from the angle’s sides or arms.
3. It divides the opposite side of a triangle into the ratio of the other two sides’ measures.

Angle Bisector Theorem:

Let’s take a deeper look at an important property of the triangle’s angle bisector, as highlighted in the previous section. The angle bisector theorem of a triangle describes this property. In a triangle, the angle bisector drawn from one vertex splits the side on which it falls in the same ratio as the other two sides of the triangle, according to the angle bisector theorem.

Statement: the triangle’s angle bisector divides the opposite side into two segments that is proportional to the other two sides of a triangle.

The Different Theorems:

1. Interior Angle Bisector Theorem: –

The angle bisector of the triangle divides the opposite side into 2 parts proportional to the other 2 sides of the triangle. The angle bisector theorem is defined as  the relative lengths of the two segments that a line bisects the opposite angle divides a triangle’s side into. It is equal to the relative lengths to the other two sides of the relative lengths of the triangle.

1. The Converse – Internal angle bisector theorem: –

In a triangle, if the interior point is equidistant from the two sides of a triangle, then that point lies on the angle bisector of the angle formed by the two-line segments.

1. Theorem of Perpendicular Bisectors: –

If a point is equidistant from the endpoints of a line segment in a triangle, it is on the perpendicular bisector of the line segment, according to this theorem. The perpendicular bisector, in other words, divides a given line segment into two equal portions to which it is perpendicular. When a perpendicular bisector is drawn from the vertex to the opposite side of a triangle, the segment is the Theorem of the external angle bisector. If a point is equidistant from the endpoints of a line segment in a triangle, it is on the perpendicular bisector of the line segment, according to this theorem. The perpendicular bisector, in other words, divides a given line segment into two equal portions to which it is perpendicular. When a perpendicular bisector is drawn from the vertex to the opposite side of a triangle, the segment is divided into two congruent segments.

1. Theorem of the external angle bisector: –

 In non-equilateral triangles, the external angle bisector halves the opposite side externally in the ratio of the sides containing the angle, and this condition is prevalently divided into two congruent segments.

Point to Remember:

• The angle bisector theorem can be defined by the relative lengths of the 2 segments that a line bisects the opposite angle divides a triangle’s side into. It is equal to their relative lengths to the other 2 sides of the triangle’s relative lengths.
• The perpendicular of a line segment is defined as a line that divides another line segment into two equal halves across its midpoint at 90 degrees. Any point on the perpendicular bisector is at an equidistant from both endpoints of the line segment on which it is drawn, according to the perpendicular bisector theorem. Let us take an example to understand this, if there is an iron pillar standing at an angle in the middle of a city’s bridge, all of the pillar’s points will be at an equal distance from the bridge’s endpoints.