An angle bisector is a ray, line, or line segment that divides a given angle into two equal (congruent) angles. It is used in geometric constructions and in studying properties of triangles and angles.
Angle Bisector of a Triangle
In a triangle, an angle bisector is a line drawn from a vertex that divides the angle into two equal parts. Each triangle has three angle bisectors, one from each vertex. These three bisectors intersect at a single point called the incenter.
The incenter lies inside the triangle and is at an equal distance from all three sides. In the figure, AG, BD, and CE are the angle bisectors, and their point of intersection F is the incenter.
Properties of Angle Bisector
- An angle bisector divides an angle into two equal angles.
- Any point on the angle bisector is equidistant from the sides of the angle.
- In a triangle, an angle bisector divides the opposite side in the ratio of the other two sides.
Construction
Steps of Construction
To construct an Angle Bisector for an angle using a ruler and compass, follow these steps:
- Draw an angle ∠ABC.
- With B as the centre, draw an arc cutting BA and BC at D and E.
- With D and E as centres and the same radius, draw arcs intersecting at F.
- Join B to F. Then, BF is the required angle bisector of ∠ABC.
Angle Bisector Theorem
In a triangle, the bisector of an angle divides the opposite side into segments that are proportional to the other two sides of the triangle.
In ΔPQR, if PS is the bisector of ∠P meeting QR at S, then:
PQ / PR = QS / SR or a / b = x / y
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Solved Example
Example 1: If an Angle Bisector divides an angle of 120 degrees, then what will be the measure of each angle?
Solution:
Given, a angle is 120 degrees
As we know, the Angle Bisector splits the angle into equal two parts.
Therefore, 120 degrees is divided into equal two parts, say z.
Hence,
z + z = 120°
2z = 120°
z = 120°/2
z= 60°
Example 2: A ray PX divides an angle PQR into two equal parts. If one part is equal to 5x – 5 and the second part is equal to 20, then what is the value of x?
Solution:
Given, PX divides angle PQR into two equal parts. Thus, PX is the angle bisector.
Now, each part should measure equal.
Thus,
5x – 5= 20
5x = 20 + 5 = 25
x = 25/5 = 5
Hence, the value of x is 5.
Example 3: A ray QT is the bisector of ∠PQR, and QU bisects ∠PQT. Find the measure of ∠TQU given that ∠PQR=120.
Solution:
It is given that ∠PQR=120°. Also, ∠PQT = 1/2 × ∠PQR = 1/2 × 120° = 60° (QT is an angle bisector bisecting ∠PQR into two equal parts) Now, ∠TQU = 1/2 × ∠PQT = 1/2 × 60° = 30° (QU is a bisector and bisects ∠PQT into two equal parts) ∴ The value of ∠TQU is 30°.
Example 4: A ray drawn from point O is the Angle Bisector of ∠POQ. If both bisector angles are 7x - 2 and 12, then find the value of x.
Solution:
Since the ray is an angle bisector, it divides the angle into two equal parts.
So,
7x − 2 = 12
7x = 12 + 2
7x = 14
x = 2∴ The value of x is 2.
Example 5: If ∠ACF = ∠FCB = 60° and CF is the angle bisector, find ∠ACB.
Solution:
We know that an Angle Bisector splits an angle into two equal sections.
Since ∠ACF = ∠FCB = 60°, we can say that ∠ACB= 60°+ 60°= 120°.
Practice Questions
Q1. If an Angle Bisector divides 280degrees, then what will be the measure of each angle?
Q2. Construct the Angle Bisector of 100 degrees angle.
Q3. Construct an Angle Bisector of 20 degrees.
Q4. If an Angle Bisector splits a right angle, what will be the measure of each angle?
Q5. Find the value of x if the ray OM is an Angle Bisector and both angles are 3x-8 and x.
Q6. If an Angle Bisector splits an angle, what would be the measurement of each angle thus formed?