Geometry for Elementary School/Bisecting an angle

BISECT ANGLE $\angle ABC$

1. Use a compass to find points D and E, equidistant from the vertex, point B.
2. Draw the line ${\overline {DE}}$.
   
   
   
   
3. Construct an equilateral triangle on ${\overline {DE}}$ with third vertex F and get $\triangle DEF$. (Lines DF and EF are equal in length).
   
   
   
   
4. Draw the line ${\overline {BF}}$.
   
   
   
   

Claim

1. The angles $\angle ABF$, $\angle FBC$ equal to half of $\angle ABC$.

The proof

1. ${\overline {DE}}$ is a segment from the center to the circumference of $\circ B,{\overline {BD}}$ and therefore equals its radius.
2. Hence, ${\overline {BE}}$ equals ${\overline {BD}}$.
3. ${\overline {DF}}$ and ${\overline {EF}}$ are sides of the equilateral triangle $\triangle DEF$.
4. Hence, ${\overline {DF}}$ equals ${\overline {EF}}$.
5. The segment ${\overline {BF}}$ equals to itself
6. Due to the Side-Side-Side congruence theorem the triangles $\triangle ABF$ and $\triangle FBC$ congruent.
7. Hence, the angles $\angle ABF$, $\angle FBC$ equal to half of $\angle ABC$.

Note

We showed a simple method to divide an angle to two. A natural question that rises is how to divide an angle into other numbers. Since Euclid's days, mathematicians looked for a method for trisecting an angle, dividing it into 3. Only after years of trials it was proven that no such method exists since such a construction is impossible, using only ruler and compass.

Exercise

1. Find a construction for dividing an angle to 4.
2. Find a construction for dividing an angle to 8.
3. For which other number you can find such constructions?