Measurement of An Angle Outside the Arc of a Circle

 

By

Nancy Kurtz

 

         The purpose of this investigation is to discover the formula for determining the measurement of an angle that lies outside the arc of a circle.  The following illustration shows the information given to begin the solution.

 

         Our problem is to determine the measure of ÐB.  If we know that ÐDAE = 1/2arc(DE) and ÐE = 1/2arc(AC), we can use those measurements to develop an equation for the solution of ÐB.   We also know that angle ÐDAE = ÐE + ÐB; therefore, ÐB=ÐDAE - ÐE.

 

         Through substitution we then write:

 

ÐB=1/2arc(DE) – 1/2arc(AC)

 

 

 

 

 

         To prove our theorem, begin by measuring angles DAE and E.

 

 

ÐB=1/2arc(DE)-1/2arc(AC)

ÐB=1/2(arcDE-arcAC)

ÐB=1/2(88.26-23.54)

ÐB=1/2 (64.72)

ÐB=32.36

 

         We then use our measuring function to determine if ÐB is correct.

 

         In conclusion, we can see that our theorem has been proven and also shown to be correct when we input the values for the known relationships.

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