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Angle Bisector

An angle bisector is a ray that cuts an angle into two equal parts. Every point on it is the same distance from both arms of the angle, and every point that is the same distance from both arms lies on it. You can draw it with only a compass and a ruler.

🎬 Step-by-step story

  1. This is an angle. Two arms meet at a corner called O. Both arms are open.
  2. Now a purple ray starts at O. It cuts the angle into two equal parts: blue and green. This ray is the angle bisector.
  3. Pick a point P on the purple ray. Drop a straight line from P to each arm, at 90 degrees. Both distances are the same.
  4. Why? Two triangles are made. Fold the green one over the purple line. It fits on the blue one exactly, so the distances match.
  5. Now push the point off the line to Q. One distance gets short, the other long. Equal distances happen only on the purple ray.
  6. Free play: change the angle, slide P along the ray, and slide it sideways. Watch when the two distances become equal.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Is a bisector a line or a ray?

It starts at the vertex and goes outward, so it is a ray. We often draw it as a line segment.

Why must the distance be at 90 degrees?

A slanted line to the arm is longer. The straight drop at 90 degrees is the shortest, and that is what distance means.

Why do the two triangles match?

They share the side OP, have equal angles at O, and both have a 90 degree angle. Folding on the bisector shows them landing on each other.

Can a point off the bisector be equally far from both arms?

No. Inside the angle, the only points with equal distances are on the bisector. Slide Q sideways to see the distances differ.

Does this work for any angle size?

Yes. Change the angle slider and the distances stay equal for points on the ray.

What is an angle bisector?

An angle is made by two arms that start at one point. That point is called the vertex.

A bisector cuts something into two equal parts. The angle bisector is a ray from the vertex that cuts the angle into two angles of the same size.

If the angle is 80°, each half is 40°. If the angle is 2x, each half is x.

Construction with ruler and compass

You do not need a protractor. Follow these steps for any angle with vertex O.

  1. Put the compass point on O. Draw an arc that cuts both arms. Call the cutting points D and E.
  2. Put the compass point on D. Draw a small arc inside the angle.
  3. Keep the same opening. Put the compass point on E. Draw another arc that crosses the first one. Call the crossing point C.
  4. Join O to C with a ruler. Ray OC is the angle bisector.

Why does it work? OD = OE (same radius) and DC = EC (same opening). OC is shared. So the triangles ODC and OEC match on all three sides, and their angles at O are equal.

The equidistance property

Distance from a point to a line always means the shortest distance, which is the straight line at 90°.

Property: Every point on the bisector of an angle is the same distance from both arms.

Proof. Let P be on the bisector of angle AOB. Draw PD ⟂ OA and PE ⟂ OB. In triangles ODP and OEP: angle POD = angle POE (bisector), angle ODP = angle OEP = 90°, and OP is common. By AAS the triangles are congruent, so PD = PE.

The converse: equal distances mean on the bisector

The converse turns the statement around: a point inside an angle that is the same distance from both arms lies on the bisector.

Proof. Let PD ⟂ OA, PE ⟂ OB and PD = PE. In the right triangles ODP and OEP, the hypotenuse OP is common and PD = PE. By RHS they are congruent, so angle POD = angle POE. Hence OP bisects the angle.

So the bisector is exactly the set of points equally far from both arms. This is why it is called a locus (see Locus).

Try it: fold and check

Draw an angle on paper and cut it out. Fold one arm exactly onto the other arm. The crease is the bisector. Mark any point on the crease. Measure its distance to each arm with a set square. Both numbers match! Then mark a point off the crease and measure again: they differ.

Key formulas and definitions

Worked examples

1. An angle measures 80°. What is each half after bisecting?

80° ÷ 2 = 40°. Each part is 40°.

2. Point P is on the bisector of angle AOB. Its distance from arm OA is 5 cm. What is its distance from OB?

P is on the bisector, so the distances are equal. Distance from OB = 5 cm.

3. A point is 4 cm from one arm of an angle and 4 cm from the other arm. Where does it lie?

Equal distances from both arms, so by the converse it lies on the angle bisector.

4. Angle ABC = 64° and BD is its bisector. Find angle ABD and angle DBC.

Each half = 64° ÷ 2 = 32°. So angle ABD = 32° and angle DBC = 32°.

5. In angle AOB = 100°, ray OC is the bisector and angle AOC = 2x + 10. Find x.

Half of 100° is 50°. So 2x + 10 = 50, 2x = 40, x = 20.

6. AD bisects angle A of a triangle. D is 3 cm from AB. AB = 10 cm and AC = 8 cm. Find the area of triangle ABC (as the sum of ABD and ACD).

D is on the bisector so it is also 3 cm from AC. Area ABD = ½ × 10 × 3 = 15 cm². Area ACD = ½ × 8 × 3 = 12 cm². Total = 27 cm².

7. Two straight roads meet at 60°. A lamp is placed on the bisector, 10 m from the junction. How far is it from each road?

Half angle = 30°. Distance = 10 × sin 30° = 10 × 0.5 = 5 m from each road.

Common mistakes

Practice quiz

1. An angle bisector divides the angle into:
2. A point on the bisector of an angle is:
3. In the compass construction, the two crossing arcs are drawn with:
4. A point inside an angle is 6 cm from each arm. It lies:
5. Which congruence rule proves PD = PE for a point P on a bisector?

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is an angle bisector in simple words?

It is a line that starts at the corner of an angle and cuts the angle into two equal parts, like the crease when you fold one arm onto the other.

How do you draw an angle bisector without a protractor?

Draw an arc from the vertex cutting both arms. From each cutting point draw arcs of the same opening so they cross. Join the vertex to the crossing point.

Why is every point on the bisector equally far from both arms?

The two small right triangles made by the perpendiculars have a common side and equal angles, so they are congruent. The matching sides, the distances, are equal.

Where this is taught

China八年级（初二）Ch.14 Congruent triangles

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