What is an Angle Bisector?
An angle bisector, also known as the bisector of an angle, is a ray or line that divides an angle into two equal angles. We can construct an angle bisector for angles starting from one degree! Suppose we want to construct a 30° angle, we first construct a 60° angle, and then bisect this angle. Each angle formed after the 60° angle is bisected is a 30° angle. Angle bisectors can be used to construct angles of different measures such as 45 degrees, 15 degrees and other angles of different measures.
What are the Properties of Angle Bisectors?
- The two arms of the angle are located at an equal distance from every point on the angle bisector.
- An angle bisector can be drawn for any angle, including acute, obtuse and right angles.
- In a triangle, the angle bisector divides the opposite side into two parts that are proportional to the other two sides of the triangle. This is also known as the angle bisector theorem.
Solved Examples on Angle Bisector
Example 1: An angle of 74 degrees is divided by an angle bisector. Determine the new angle formed on either side of this bisector.
Solution:
The value of the angle is 74 degrees.
As we know, the angle bisector divides the angle into equal two parts.
Therefore, 74 degrees is divided into two equal parts.
So, to find the value of the angles on either side we divide 74 into two.
\(\frac{74^{\circ}}{2}=37^{\circ}\)
Therefore the value of each angle divided by the angle bisector is 37°.
Example 2: Two logs of wood, A and B, were placed at an angle of 60 degrees as shown below. A log of wood C was placed such that the angle it makes between one of the logs is equal to 30 degrees. Find the angle made by the remaining portion and verify if the new log of wood that was inserted acts as an angle bisector.
Solution:
The total value of the angle is 60 degrees. The log that is inserted between the original two logs makes an angle of 30 degrees, that is, the angle between log A and C is 30°.
Therefore, the angle formed by the log and the remaining portion, that is, the angle between log C and B is = 60 °– 30 ° = 30 °
Since log C forms 30° on either side, log C acts as an angle bisector.
Example 3: In the figure below, a ray QX divides an angle PQR into two equal angles. If one section is represented by 3x – 8 and the second section is equal to 30°, then what is the value of x?
Solution:
The ray QX divides the angle PQR into two equal parts. This means that QX is an angle bisector.
Since both the parts are equal, we can equate one to the other.
3x – 9 = 30
3x – 9 + 9 = 30 + 9 [Add 9 on both sides of the equation]
3x = 39 [Add]
x = \(\frac{39}{3}\) = 13 [Divide]
Hence, the value of x is 13.
An angle bisector is a ray that divides an angle into two equal angles.
The attributes of an angle bisector are:
- Any point on the bisector of an angle is at an equal distance from the arms of the angle.
- In a triangle, the angle bisector divides the opposite side into two parts that are proportional to the other two sides of the triangle.
When an angle is divided into three equal parts, then the angle is said to be trisected and the procedure is called the trisection of an angle.
Angle bisectors divide the angle into two equal halves. Therefore, an angle bisector of a 180° angle will form two angles of 90° each. Such an angle bisector is called a perpendicular bisector. A 180° angle also indicates a straight line. Thus, a perpendicular bisector is a line that divides another line segment into two equal halves such that the angle formed between them is 90°.