# Geometry for Elementary School/Bisecting an angle

 Geometry for Elementary School Why are the constructions not correct? Bisecting an angle Bisecting a segment

BISECT ANGLE ${\displaystyle \angle ABC}$

1. Use a compass to find points D and E, equidistant from the vertex, point B.
2. Draw the line ${\displaystyle {\overline {DE}}}$.

3. Construct an equilateral triangle on ${\displaystyle {\overline {DE}}}$ with third vertex F and get ${\displaystyle \triangle DEF}$. (Lines DF and EF are equal in length).

4. Draw the line ${\displaystyle {\overline {BF}}}$.

## Claim

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

## The proof

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

## 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?