Geometry for Elementary School/Bisecting an angle

From Wikibooks, open books for an open world
Jump to navigation Jump to search
Geometry for Elementary School
Why are the constructions not correct? Bisecting an angle Bisecting a segment


  1. Use a compass to find points D and E, equidistant from the vertex, point B.
  2. Draw the line .
    Geom bisect angle 04.png

  3. Construct an equilateral triangle on with third vertex F and get . (Lines DF and EF are equal in length).
    Geom bisect angle 05.png

  4. Draw the line .
    Geom bisect angle 06.png

Claim[edit | edit source]

  1. The angles , equal to half of .

The proof[edit | edit source]

  1. is a segment from the center to the circumference of and therefore equals its radius.
  2. Hence, equals .
  3. and are sides of the equilateral triangle .
  4. Hence, equals .
  5. The segment equals to itself
  6. Due to the Side-Side-Side congruence theorem the triangles and congruent.
  7. 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]

  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?