Geometry for Elementary School/Bisecting an angle

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

BISECT ANGLE

  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]

  1. The angles , equal to half of .

The proof[edit]

  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]

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]

  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?