Quadrilaterals Inscribed in a Circle

By

Scottie Benford

 

 

 

 

Today in class we talked about quadrilaterals being inscribed in a circle.  The question that was posed was to answer if the opposite angles of the quadrilateral are supplementary.  I’m going to prove that by using GSP.  First I will use GSP to draw the Quadrilateral inscribed in the circle.

 

 

 

 

 

 

 

 

 

 

 

Next I will place four points on my quadrilateral using GSP.

 

 

 

 

 

 

 

I will measure two opposite angles using GSP.

 

I now have enough information to say that alternate angles of a quadrilateral inscribed in a circle are supplementary.  Now I want to use previous information to show that this is true.  Earlier in class we discussed that inscribed angles are equal to one-half the arc measure.  If my angle sum equals 180 degrees then that means that the arc that they are associated with equal 360 degrees.  GSP will be used to show the arcs that are associated with each circle.

 

 

 

 

 

 

The arc that is highlighted in red is associated with angle B.

 

 

 

 

 

 

The arc shown in yellow is the arc associated with inscribed angle ADC.  The sum of the yellow and red arc equal 360degrees, thus proving that the alternate angles of a quadrilateral inscribed in a circle are supplementary.  We must keep in mind that the inscribed angle measures are equal to half of the measure of the arcs.

 

 

 

To try this construction for yourself go to my GSP file.