Angle bisector

An angle bisector is the ray that cuts a given angle into two equal angles. It is the angle world's analogue of the perpendicular bisector , and just as easy to construct.

Concept

What we want to build

Given an angle $\angle BAC$, we want a ray $\overrightarrow{AD}$ from the vertex $A$ such that $\angle BAD = \angle DAC$ (each equal to half of $\angle BAC$).

The construction

1. Place the compass tip at $A$. Choose any convenient radius.
2. Draw an arc that crosses both arms of the angle. Call those crossings $P$ (on $\overrightarrow{AB}$) and $Q$ (on $\overrightarrow{AC}$).
3. Without changing the compass, place the tip at $P$ and draw an arc inside the angle.
4. Now move the tip to $Q$ (same compass spread) and draw another arc inside the angle.
5. The two arcs intersect at a point $D$.
6. Draw the ray $\overrightarrow{AD}$. That is the angle bisector.

Why does it work?

The point $D$ is the same distance from $\overrightarrow{AB}$ as from $\overrightarrow{AC}$. We can see this because the arcs are constructed with equal radii: $PD = QD$ and $AP = AQ$. Triangles $APD$ and $AQD$ are congruent (by side-side-side), so $\angle PAD = \angle QAD$. That is exactly what we wanted , the angle is bisected.

In words: a point is on the bisector of an angle if and only if it is equidistant from the two arms. This is the defining property.

Useful properties

• The angle bisectors of the three angles of a triangle all meet at one point, called the incentre. The incentre is the centre of the largest circle that fits inside the triangle (the incircle).
• Bisecting twice gives a fourth of the angle. Bisecting three times gives an eighth. You can keep going to chop an angle into halves of halves of halves $\dots$ but you cannot, with just ruler and compass, divide any angle into three equal parts in general. ("Trisecting an angle" was a famous unsolved problem for over $2{,}000$ years until it was proved impossible in the 19th century.)
• An angle bisector together with the angle's two arms creates two congruent triangles when you reflect across the bisector. The bisector is a line of symmetry of the angle.

Worked examples

Example 1. Bisect an angle of $80°$.

• Draw $\angle BAC = 80°$ using a protractor.
• Place compass at $A$, draw an arc cutting both arms at $P$ and $Q$.
• From $P$ and $Q$ (same spread), draw two arcs that cross at $D$.
• Draw $\overrightarrow{AD}$.
• Each new angle is $40°$. Verify with protractor.

Example 2. Bisect a right angle.

• Right angle = $90°$. After bisecting, each half is $45°$.

Example 3. Bisect a $60°$ angle, then bisect one of the halves.

• $60° \to 30°$ (first bisection).
• $30° \to 15°$ (second bisection of one half).
• The original is now split into $15°$ and $15°$ and $30°$ , total still $60°$.

Example 4. Using the construction, divide a $90°$ angle into four equal parts.

• Bisect to get $45°$ and $45°$.
• Bisect each half to get $22.5°$ four times.

Try it yourself

1. Draw an angle of $50°$ and bisect it.
2. Draw an angle of $120°$ and bisect it.
3. Bisect a straight angle ($180°$). What is each half called?
4. Draw an angle and bisect it twice (bisect, then bisect one half). Each new piece is what fraction of the original?
5. Construct a triangle and find the bisectors of all three angles. Do they meet?
6. Construct an angle of $30°$ by starting with an equilateral triangle's $60°$ angle and bisecting.
7. Construct an angle of $15°$. (Hint: bisect $30°$.)
8. Investigate: with ruler and compass, can you divide any given angle into $4$ equal parts? Into $8$? Into $3$? (Two of these are yes, one is famously no.)

Activity

Triangle's heart. Draw a triangle on a fresh sheet of paper. Bisect each of its three angles using the construction above. The three bisectors should meet at a single point inside the triangle. That point is the incentre. Now place the compass tip on the incentre and find the largest circle that fits inside the triangle without crossing any side. That is the incircle.