PROBLEM: Angle Bisector

Imagine a sheet of paper with two line segments drawn on it, as shown. Without extending the paper or the line segments, construct the bisector of the angle determined by the two line segments.

How many different solutions can you find?

The first thing is to draw any segment between L_{1} and L_{2}
such that the endpoints of this segement line-on L_{1} and L_{2}.
This is illustrated in the diagram below.

We now have two vertices of a triangle and if we knew where the third vertice was we could construct the angle bisector. The reason for drawing the red segment is because we want to create an isosceles triangle and we need to know what the measure of the base angles needs to be.

Next, copy the angle formed by the red segment and L_{2} and
add it to the angle formed by the red segment and L_{1}. Now, we
can draw a new segment that represents the total angle. This step is shown
in the figure below. The blue segment represents this new segment.

The angle marked in red is the angle that was copied to the position
of the angle marked in blue. The obtuse angle formed by L_{1} and
the blue segment has the measure of the sum of the two base angles of a
triangle. If this triangle is isosceles, then each of the base angles has
the same measure. So, the next step will be to bisect this angle and construct
the two base angles of an isosceles triangle. The reason we want to construct
an isosceles triangle is because we know that the angle bisector of the
one angle that is not a base angle of an isosceles triangle is a perpendicular
bisector of the side included by the base angles. Therefore, once we have
the base angles for an isosceles triangle, we can construct the side included
by them and then construct the perpendicular bisector of this side and we
will have the angle bisector.

The figures below illustrate bisecting the obtuse angle formed by L_{1}
and the blue segment, copying this angle to both L_{1} and L_{2}
and constucting the segment included by these two new angles.

Figure 1` |
Figure 2 |

The green ray in **Figure 1** above resembles the angle bisector of
the angle marked in yellow. **Figure 2** illustrates copying the bisected
angle to both L_{1} and L_{2} and then constructing the
segment from the point of intersection of the L_{1} and the green
ray and the point of intersection of L_{2} and the green ray. Now,
we have the side included by the base angles of an isosceles triangle (the
solid green segment in **Figure 2**). All that we need to do now is find
the perpendicular bisector of this green segment and we will have the angle
bisector of the original two segments that we were given. This perpendicular
bisector is shown in the figure below.

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©1998 by Luke Rapley