Geometry for Elementary School/Bisecting an angle

Template:Navigate BISECT ANGLE $\mathrm{\angle }ABC$

1. Use a compass to find points D and E, equidistant from the vertex, point B.
2. [[../Our_tools:_Ruler_and_compasses# how to draw a line?| Draw the line]] $\overline{DE}$.
   
   
   
   
3. [[../Constructing equilateral triangle|Construct an equilateral triangle]] on $\overline{DE}$ with third vertex F and get $\mathrm{△}DEF$. (Lines DF and EF are equal in length).
   
   
   
   
4. [[../Our_tools:_Ruler_and_compasses# how to draw a line?| Draw the line]] $\overline{BF}$.
   
   
   
   

Claim

1. The angles $\mathrm{\angle }ABF$, $\mathrm{\angle }FBC$ equal to half of $\mathrm{\angle }ABC$.

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 $\mathrm{△}DEF$.
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 Side-Side-Side congruence theorem]] the triangles $\mathrm{△}ABF$ and $\mathrm{△}FBC$ congruent.
7. Hence, the angles $\mathrm{\angle }ABF$, $\mathrm{\angle }FBC$ equal to half of $\mathrm{\angle }ABC$.

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 [[../Some impossible constructions#Trisecting the angle|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?