An angle bisector is defined as a ray that divides a given angle into two angles with equal measures. The word bisector or bisection means dividing one thing into two equal parts. In geometry, we usually divide a triangle and an angle by a line or ray which is considered as an angle bisector.
|1.||Properties of an Angle Bisector|
|2.||How to Construct an Angle Bisector?|
|3.||Angle Bisector Theorem|
|4.||Solved Examples on Angle Bisector|
|5.||Practice Questions on Angle Bisector|
|5.||FAQs on Angle Bisector|
Properties of an Angle Bisector
In geometry, we read about many types of angles. Each angle has its own property. Bisecting an angle gives a particular angle different characteristics. Following are the two major properties that an angle bisector holds.
1) Any point on the bisector of an angle is equidistant from the sides of the angle.
2) In a triangle, the angle bisector divides the opposite side in the ratio of the adjacent sides.
How to Construct an Angle Bisector?
Let's try constructing the angle bisector for an angle. In this section, we will see the steps to be followed for angle bisector construction. Let's begin!
Steps to Construct an Angle Bisector
Step 1: Draw any angle, say ∠ABC.
Step 2: Taking B as the center and any appropriate radius, draw an arc to intersect the rays BA and BC at, say, E and D respectively. (Refer to the figure below)
Step 3: Now, taking D and E as centers and with a radius more than half of DE, draw an arc to intersect each other at F.
Step 4: Draw ray BF. This ray BF is the required angle bisector of angle ABC.
Angle Bisector Theorem
Let's now understand in detail an important property of the angle bisector of a triangle as stated in the previous section and then solve angle bisector problems. This property is known as the Angle Bisector Property of 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.
Given : Δ ABC; AD bisects ∠BAC
To Prove: AB/AC=BD/DC. We will prove this result using properties of parallel lines and similarity of triangles.
Construction: We construct an auxiliary line passing through C and parallel to side AB, intersecting AD extended at point E.
- Step 1: As AB||CE, ∠1=∠3 (∵ alternate angles)
- Step 2: But, ∠1 = ∠2 (∵ AD is angle bisector)
- Step 3: From, Steps (1) and (2) we get, ∠2=∠3 ⇒Δ ACE is an isosceles triangle ⇒x=y
- Step 4: Now, ΔABD∼ΔECD (ByAA test of similarity) ∴AB/EC = BD/CD (corresponding angle of similar triangles)(corresponding angle of similar triangles) p/y = a/b
- Step 5: So from step 4, we got p/y = a/b. Also from step 3, we have x = y ⇒ p/x = a/b i.e., AB/AC = BD/DC
Related Articles on Angle Bisector
Read below mention interesting articles to learn more about the angle bisector.
Solved Examples on Angle Bisector
Example 1. In the figure given below, BD is the bisector of ∠ABC and BE bisects ∠ABD. Find the measure of ∠DBE given that ∠ABC=80°.
It is given that ∠ABC=80°.
∠ABD = 1/2 × ∠ABC = 1/2 × 80° = 40° (BD is an angle bisector bisecting ∠ABC in two equal parts)
Now,∠DBE = 1/2 × ∠ABD = 12 × 40° = 20° (BE is a bisector and bisects ∠ABD in two equal parts)
∴ ∠DBE = 20°
Example 2: In the figure, Ray OD is the angle bisector. Find x.
To find x, we will be using the property: Any point on the bisector of an angle is equidistant from the sides of the angle.
So, the point D which lies on the bisector OD will be equidistant from sides OB and ON ⇒ DB = DN ⇒ 3x − 2 = 10
⇒ 3x = 2 + 10 ⇒ 3x = 12. ∴ x = 4
Example 3: If QS is the bisector of ∠PQR, find x.
As QS bisects ∠PQR, by angle bisector theorem we get, QP/QR = PS/RS ⇒ 18/24 = 12/x ⇒ x = (12 × 24)/18
⇒ (2 × 24)/3 =2 × 8 = 16. ∴ x = 16
FAQs on Angle Bisector
What are the Properties of Angle Bisector?
An angle bisector has two properties:
- Any point on the bisector of an angle is equidistant from the sides of the angle.
- In a triangle, the angle bisector divides the opposite side in the ratio of the adjacent sides.
Does Angle Bisector Cut an Angle in Half?
Yes, an angle bisector divides the given angle into two equal angles. In other words, we can say that the measure of these angles will be half of the original angle.
What is the Property of Angle Bisector of Triangle?
The property of angle bisector of triangle states that the angle bisector divides the opposite side of a triangle in the ratio of the adjacent sides of it.
What is an Angle Bisector and What Does it Do?
When a line or ray divides an angle into two congruent angles, it is known as the angle bisector.
Which is the Best Definition for Angle Bisector?
An angle bisector is nothing but a ray that divides an angle into two congruent parts. The ray is considered as an angle bisector.
Does the Angle Bisector go Through the Midpoint?
Can an Angle Have More Than One Angle Bisector?
No, an angle can have only one angle bisector. For example, if we bisect 60° angle we will get 30° as a result. This means 60° angle is divided into two equal angles (30° each). Hence 60° angle can only be bisected once. Further, we can again bisect 30° angle into two equal angles as (15° each).