# Geometry for Elementary School/Bisecting an angle

Jump to navigation
Jump to search

**BISECT ANGLE**

- Use a compass to find points D and E, equidistant from the vertex, point B.
- Draw the line .
- Construct an equilateral triangle on with third vertex
**F**and get . (Lines DF and EF are equal in length). - Draw the line .

## Claim

[edit | edit source]- The angles , equal to half of .

## The proof

[edit | edit source]- is a segment from the center to the circumference of and therefore equals its radius.
- Hence, equals .
- and are sides of the equilateral triangle .
- Hence, equals .
- The segment equals to itself
- Due to the Side-Side-Side congruence theorem the triangles and congruent.
- Hence, the angles , equal to half of .

## Note

[edit | edit source]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

[edit | edit source]- Find a construction for dividing an angle to 4.
- Find a construction for dividing an angle to 8.
- For which other number you can find such constructions?