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# Angle Bisector|Definition & Meaning

## Definition

An** angle bisector **is a line that cuts an **angle **made by two **adjoining** lines **equally**. This means that the angle is made by two joining lines or made up of three points **A, B, and C**, with the angle between **line AB** and **line BC**, cut in **half** by an **angle bisector DB,** where C is a point in the graph.

In general, a **bisector** that bisects means to** cut into two halves**. Here, we are cutting an angle into two halves by using an **angle bisector** crossing that angle through the **center point **(in the above case, the line BD is the angle bisector).

These bisectors are very handy in making **smaller angles** from **larger angles **that are already given. Let us say that we are given an **angle of 60°,** and we ideally want a smaller angle of **30°**. All that we are required to do is draw that 60° angle between two lines and then draw an angle bisector between the two. This gives us** two 30° **angles to work around with. There are even further applications of an angle bisector that will be discussed later in this article.

## Properties of an Angle Bisector

An angle bisector is usually renowned for its **properties **that aid in the easy construction and organization of shapes and structures in many **real-life applications**. Firstly, an angle bisector can bisect any angle ranging from **0° to 360° **(that is a whole** circle**). This means that **any angle **can be **bisected **into equal halves by an angle bisector.

Furthermore, the **points **on the angle bisector are all **equidistant** to the** two adjoining line segments** of the respective bisected angle. This gives us a **symmetrical shape** and the two sides are mirror images of each other, which further **aids in the geometrical constructions** of many shapes.

Moreover, considering an angle bisector of one of the points of a triangle, it will cut the **opposite side of the triangle** into a** ratio** based on the **ratio of arm length** of the other adjacent sides of the bisecting angle point. This means that the bisecting line will not always cut the triangle in half and the bisected figure will **not be symmetrical**.

Additionally, in a triangle, taking the angle bisector on all **three vertices **of a triangle results in an** intersecting point** on the triangle. This intersection denotes the** center of the triangle** and is called the **incenter.**

Similarly, to find a **center of a circle geometrically,** we can draw three **tangential **adjoined lines to make a triangle around it. Thus, by taking the three angle bisectors of the triangle, the intersecting point is the center of the circle.

## Construction of an Angle Bisector

In today’s era of technological advancements, one can get his hand on any **angle bisector constructing tool **on the internet, which makes it easy for the common person to bisect any angle in half. Nevertheless, the **geometrical construction** on a piece of paper is very simple as well. All you need is **graphing paper, a compass, a ruler, and a lead pencil**.

Let us say we are given an **angle ∠ABC **made between the** two adjoining line** segments **AB **and **BC**. If needed, this angle can be measured using a **protractor** and can be **halved **as well using. But in real-life cases where we might not have a protractor in our hands at all times, we can use the compass to find its half.

Firstly, we open the **compass **at a** fixed length for all times,** and from the origin point, we draw two **arcs** that intersect the two line segments individually. After that, we place our pivoting arm of the compass on the intersection of one of the line segments and the arc to draw an arc towards the inside of the line segment. A similar procedure is followed for the other line segment.

These arcs will intersect with one another and we can call this point **D**. Afterwards, we draw a **line BD**, using a ruler, between points **B** and **D**. This line is the resulting angle bisector that will split the angle **∠ABC** into half. Moreover, we have to place great emphasis on the opening of the compass which has to be kept the same throughout the construction of the angle bisector.

## Significance of the Properties of an Angle Bisector in Its Construction

This construction of the angle bisector **utilizes one of the properties** of the angle bisector which is **equidistant **from the two line segment at all points. Hence, the same points on the two line segments of a **specific distance **from the **pivot point** always have the same distance from the angle bisector.

Furthermore, the property that an angle bisector can be made for **any angle is also true** during its **geometric construction** using a compass where when we are given two line segments, it is always possible to construct an angle bisector regardless of the angle between the two adjoining line segment.

Moreover, we can use the angle bisector theorem to find the **length of the opposite side** of the bisecting angle that is cut into two parts using the ratio of the adjacent sides of the angle.

In **Figure 4****,** we can observe that the **ratio **between **lines AB **and **BC** is equal to the **ratio **between** line AD **and **DC**. This is known as the **angle bisector theorem** that is written as $\frac{AB}{BC} = \frac{AD}{DC}$

**Further Explanation of an Angle Bisector Theorem Using an Example**

Let us consider **figure 4**. We have a length of line **AB = 10.3115** and the length of line** BC = 4.5995.** We are given a length of the line **AD = 5.6505.** **Prove** that the** length** of the **line DC** is equal to** 2.5204** using the **angle bisector theorem. **

### Solution

Using the **angle bisector theorem,** we can find the length of **AD** by inputting the above-given values in the following equation:

\[ \frac{AB}{BC} = \frac{AD}{DC} \]

\[ \frac{10.3115}{4.5995} = \frac{5.6505}{DC} \]

\[ \frac{(5.6505)(4.5995)}{10.3115} = DC \]

\[ \mathbf{DC = 2.5204}\]

Hence proven that the angle bisector theorem proves the above values are **valid**.

*All mathematical drawings and images were created with GeoGebra.*