Geometry for Elementary School/ Bisecting an angle

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Contents

[edit] Introduction

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}.
    Geom bisect angle 04.png


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


  4. Draw the line \overline{BF}.
    Geom bisect angle 06.png


[edit] Claim

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

[edit] 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 .

[edit] 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.

[edit] 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?
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