Measurements of Inscribed Angles

By

Ricky Joe Hackett

June 3, 2003

Consider a circle that contains two lines passing through a point B located somewhere inside a circle (Figure 1).

The problem to be discussed
is the mÐADB
and the mÐCBE
expressed as a measurement of the intercepted arcs.

First, draw a line segment
from point A to point E as illustrated in Figure 2.

Figure 2

From previous discussions, we
know that the mÐCBE and the mÐDBA are congruent since they are vertical angles. Previously, it has also been shown that
when the vertex of an inscribed angle is located on the circumference of the
circle, the measurement of the angle is 1/2 the arc measurement created by the
sides of the angle.

By drawing the chord AE, an
inscribed angle lying on the circumference has been created. Also, the mÐCBE = mÐBAE + mÐBEA since ÐCBE is an exterior angle to DABE.

To recap:

mÐCBE @ mÐDBA

mÐCBE = mÐCAE + mÐDEA

mÐDEA = 1/2 arc measure of AD

mÐCAE = 1/2 arc measure of CE

Substituting:

mÐCBE = 1/2 arc measure of AD + 1/2 arc measure of CE

mÐCBE = 1/2 (arc measure of AD + arc measure of CE)

Likewise
mÐDBA can be shown

These relationships can be shown using GSP as demonstrated in Figure 3.

Figure 3

Figure
3 can also be viewed with GSP by clicking here.