Measurements of Inscribed Angles




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:



mÐDEA = 1/2 arc measure of AD

mÐCAE = 1/2 arc measure of CE



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.