8/10/2019 basic principle periapical.docx

1/2

The bisecting-the-angle or bisecting angle technique is based on the principle of aiming the

central ray of the x-ray beam at right angles to an imaginary line which bisects the angle formed

by the longitudinal axis of the tooth and the plane of the film packet. While it is not necessary to

go into a long dissertation on plane geometry to understand this concept, a quick review will helpmake the technique more clear. To bisect is to divide a line or angle into two equal portions. A

bisector is a plane or line that divides a line or angle into two equal portions. (Figure 37) showsan equilateral triangle, with legs AB=BC=CA, and the angles ABC=60 degrees, CAB=60degrees and BCA=60 degrees.

Figure 37.

We see (in Figure 37) the following:

1. The dotted line BD bisects the triangle, dividing it exactly in half. Thus, two equal

triangles are formed from the original. Legs AB and BC were unchanged and thus arestill equal.

2. The original line CA was divided in half by D, and thus the lines AD and CD are equal.

3. We know that the angle at point B was 60 degrees, and since it was bisected (divided

equally), it now is 30 degrees at the intersections of AD and BD.

4. We also know that bisecting the angle did not affect the angle at the old point A which

was 60 degrees, and still is.

5.

The angle at the bisecting point DC must be 90 degrees because the sum of all the anglesin any triangle is 180 degrees, and thus 180-(60+30)=90.

6. Cyzynskis Rule of Isometry states that two triangles are equal when they share onecomplete side, and have two equal angles. We can see that triangles ADB and BDC

share the common side BD.

7. We know further that the angles ADB and BDC are equal because D was defined as a

bisector of the old angle ABC.

8/10/2019 basic principle periapical.docx

2/2

8. Lastly, we know that the angles CAB and BCA were unchanged by bisecting and are still

equal. Therefore, under Cyzynskis Theorem, we can prove the triangles ABD and CBD

are equal.

In dental radiography, the theorem is applied in the following manner. The film is positioned

resting on the palate or on the floor of the mouth as close to the lingual tooth surfaces aspossible. The plane of the film and the long (vertical) axis of the teeth to be radiographed form

an angle with the apex at the point where the film packet contacts the teeth. The apex in (Figure

38) is located at the point labeled B.

In (Figure 38), the long axis through the tooth forms one leg of a triangle (AB), the plane of the

film packet another leg, (BC), both of which intersect at the apex, point B. A line representingthe central x-ray beam will form the third leg of the triangle, AC. If an imaginary line bisected

this axis-packet- ray triangle, the bisector, DB, would form the common side of two equal

triangles as defined by Cyzynskis Theorem.

Figure 38.

Since the sides formed by the tooths long axis and the film packet are equal, the image cast ontothe radiographic film would be the same length as the tooth or teeth casting that image. Thislinear equality is the basis for diagnostic quality bisecting angle radiographs.