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:

1. Put the compass point on the vertex B and draw an arc that crosses both arms, at points D and E.
2. Without changing the compass width, put the point on D and draw an arc in the interior of the angle.
3. Keeping the same width, put the point on E and draw a second arc that crosses the first at a point F.
4. 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.