How to Construct a Bisector of a Given Angle

In geometry, an angle bisector is a line that divides an angle into two equal angles. The term “bisector” refers to something that divides a shape or object into two equal halves. An angle bisector is defined as a ray that divides a given angle into two equal pieces of the same measure.

Let us quickly review the different sorts of angles before discussing an angle bisector. An angle can be acute (less than 90 degrees, such as a 60-degree angle), obtuse (greater than 90 degrees), or right depending on the inclination between the two arms (exactly 90-degrees). Angle building is an important component of geometry because it may be applied to the creation of other geometric shapes, particularly triangles. By simply bisecting several common angles, a variety of angles can be created.

What is Angle Bisector?

A bisector of an angle, also known as an angle bisector, is a ray that divides an angle into two equal pieces. For example, if ray AX cuts a 60-degree angle into two equal parts, each measurement will be 30 degrees.

An angle bisector exists for every angle. It is also the symmetry line that connects the two arms of an angle, allowing you to build smaller angles using it. Let’s say you have to build a 30° angle. By making a 60° angle and then bisecting it, you can accomplish this. Similarly, an angle bisector is used to generate 90-degree, 45-degree, 15-degree, and other angles.

Angle Bisector of a Triangle

One angle bisector is a straight line that divides an angle into two equal or congruent angles in a triangle. Every triangle can have three angle bisectors, one for each vertex. The incenter of a triangle is the place where these three angle bisectors meet. The distance between the incenter and all of a triangle’s vertices is the same. Take a look at the graphic below, which depicts a triangle’s angle bisector. The angle bisectors of BAC, ACB, and ABC, respectively, are AG, CE, and BD. F is the incenter, or point of intersection, of all three bisectors, and it is located at an equal distance from each vertex.

Properties of an Angle Bisector

By now you may have an idea about what angle bisector means in geometry. Let us look at some of the qualities of the angle bisector, which are stated below:

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

Construction of Angle Bisector

Let us try constructing the angle bisector for an angle. In this section, we will see the steps to be followed for angle bisector construction.

Steps to Construct an Angle Bisector:

1st Step: Draw any angle you want, such as ABC.
2nd Step: Draw an arc with B as the centre and any relevant radius to cross the rays BA and BC at, say, E and D. (See the diagram below.)
3rd Step: Draw two arcs to intersect each other at F, using D and E as centres and the same radius as in the previous step.
4th Step: Make a ray by connecting B and F. The needed angle bisector of angle ABC is ray BF.

Angle Bisector Theorem

Let us now understand in detail an important property of the angle bisector of a triangle as stated in the previous section. This property is known as the angle bisector theorem of a triangle. According to the angle bisector theorem, in a triangle, the angle bisector drawn from one vertex divides the side on which it falls in the same ratio as the ratio of the other two sides of the triangle.

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

PS is the angle bisector of P in PQR in the figure above. As a result, we can assert that PQ/PR = QS/SR or a/b = x/y using the angle bisector theorem.

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