Angle Bisector: Properties & Construction
Book A Free Math Class
Angle Bisector: Properties & Construction
TL;DR
An angle bisector is a ray, line, or segment that divides an angle into two equal smaller angles. This article covers the definition, the key properties, the compass-and-straightedge construction, the angle bisector of a triangle and the incenter, and six worked examples.
What Is an Angle Bisector?
An angle bisector is a ray (or line or segment) that passes through the vertex of an angle and divides it into two angles of equal measure. If ray B bisects ∠ABC, then:
∠ABD=∠DBC=12∠ABC.
The word bisect means "cut into two." So bisecting a 60° angle gives two 30° angles; bisecting a right angle (90°) gives two 45° angles. The bisector must pass through the vertex — a line that splits the opening but misses the corner is not a bisector.
Properties of an Angle Bisector
A few properties do most of the work in problems, and each one is worth stating on its own line.
- An angle has exactly one bisector. There is only one ray from the vertex that cuts the angle into two equal halves.
- Every point on the bisector is equidistant from the two arms. Drop a perpendicular from any point on the bisector to each arm; the two perpendicular distances are equal.
- The bisector works for any angle type. Acute, right, or obtuse — every angle has a bisector that halves it.
- In a triangle, an angle bisector divides the opposite side in a fixed ratio. This is the angle bisector theorem.
A reader question that comes up often: does the angle bisector always pass through the midpoint of the opposite side? No. In a triangle it generally does not. The midpoint line is the median, a different cevian.
How to Construct an Angle Bisector
The classic construction bisects any angle using only a compass and straightedge. To bisect ∠ABC:
- Put the compass point on the vertex B and draw an arc that crosses both arms, at points D and E.
- Without changing the compass width, put the point on D and draw an arc in the interior of the angle.
- Keeping the same width, put the point on E and draw a second arc that crosses the first at a point F.
- Draw a ray from B through F. Ray BF is the bisector of ∠ABC.
The Angle Bisectors of a Triangle and the Incenter
A triangle has three interior angles, so it has three angle bisectors, which meet at a single point, called the incenter.
The incenter sits at the equidistant property's natural conclusion. Each bisector is the set of points equidistant from two of the triangle's sides; the point on all three bisectors is therefore equidistant from all three sides. That equal distance is the radius of the incircle.
Angle Bisector vs Perpendicular Bisector
The two get confused because both contain the word "bisector," but they cut different things.
| Feature | Angle bisector | Perpendicular bisector |
|---|---|---|
| What it cuts in half | An angle | A line segment |
| Passes through | The angle's vertex | The segment's midpoint, at 90° |
| Equidistant from | The two arms of the angle | The two endpoints of the segment |
| In a triangle, the three meet at | The incenter | The circumcenter |
Examples of Angle Bisector
Example 1 - Ray B bisects ∠ABC, and ∠ABC=76°. Find ∠ABD.
The bisector halves the angle: ∠ABD=12(76°)=38°.
Example 2 - Ray QS bisects ∠PQR. ∠PQS=(3x+5)° and ∠SQR=(5x−15)°. Find x and ∠PQR.
Set the halves equal: (3x + 5) = (5x - 15), giving x = 10. Thus, ∠PQR = 35° + 35° = 70°.
Example 3 - A point P lies on the bisector of ∠ABC. Its perpendicular distance to arm BA is 6 cm. What is its distance to arm BC?
The distance to BC is also 6 cm.
Example 4 - Ray BF bisects ∠ABC, and one half ∠ABF=(2y+12)° while the whole angle ∠ABC=(5y−6)°. Find y.
Setting 2(2y + 12) = 5y - 6 gives y = 30.
Example 5. In triangle ABC, where do all three angle bisectors meet?
They meet at the incenter.
Example 6 - A right angle (90°) is bisected again. What is the smallest angle produced?
The smallest angle produced is 22.5°.
Key Takeaways
- An angle bisector splits an angle into two equal halves.
- Every point on the bisector is equidistant from the two arms.
- A triangle's three angle bisectors meet at the incenter, the centre of the inscribed circle.
Frequently Asked Questions
What is an angle bisector? A ray, line, or segment that passes through an angle's vertex and divides it into two equal smaller angles.
How many bisectors does an angle have? Exactly one.
Where do the three angle bisectors of a triangle meet? At the incenter, equidistant from all sides.