# 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 $\angle ABC$ 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?