Bisect - cut into two congruent (equal) pieces.
(It may be advantageous to instruct students in the use of the "large arc method" because it creates a "crayfish" looking creature which students easily remember and which reinforces the circle concept needed in the explanation of the construction.)
To understand the explanation you will need to label the point of
intersection of the arcs above segment
as D and below segment
E. Draw segments
. All four of these segments
are of the same length since they are radii of two congruent circles. More specifically, DA = DB and EA = EB.
Now, remember a locus theorem: The locus of points equidistant from
two points, is the perpendicular bisector of the line segment determined
by the two points. Hence,
is the perpendicular
bisector of .
5. Without changing the width of the compass, place the point of the compass on the other intersection point on the side of the angle and make the same arc. Your two small arcs in the interior of the angle should be crossing.
6. Connect the point where the two small arcs cross to the vertex A of the angle.
You have now created two new angles that are of equal measure (and are each 1/2 the measure of .)
Explanation of construction: To understand the explanation, some additional labeling will be needed. Label the point where the arc crosses side as D. Label the point where the arc crosses side as E. And label the intersection of the two small arcs in the interior as F. Draw segments and . By the construction, AD = AE (radii of same circle) and DF = EF (arcs of equal length). Of course AF = AF. All of these sets of equal length segments are also congruent. We have congruent triangles by SSS. Since the triangles are congruent, any of their leftover corresponding parts are congruent which makes equal (or congruent) to .