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).

 

 

 

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.