Measuring Outside Angles Using Arcs

By:

Allison Kootsikas


What is the relationship between the angles formed by two secants which intersect outside the circle and the arcs that they intersect?


 

First, I start by using GSP to construct a circle and the two secants.

The next step is to draw in two secants to form triangles.  I also highlight these triangles to show the distinct triangles.

Now, I set out to find my task.  I want to find out what ÐAEC is.  I know that ÐCAD is 1/2 of the measure of arc CD.  This means that ÐEAC is 180°-1/2 arc CD. 

 

I also know that ÐBCA is 1/2 the measure of arc BA. 

 

The sum in the interior angles of a triangle is 180°. 

ÐAEC=180°-ÐEAC-ÐECA

         =180°-(180°-arc CD/2)-(arc BA/2)

         =(arc CD/2)-(arc BA/2)

         =1/2(arc CD-arc BA)

 

Here is a copy of my GSP sketch.