Geometry Proof - The Inscribed Angle is 1/2 of the Central Angle

Date: 2/12/96 at 12:25:59
From: Anonymous
Subject: Geometry Question

I am in grade 7 but I have passed the grade 7 and 8 courses and am 
just doing all the unit tests in the grade 9 book before taking that 
exam.

The grade 9 math book describes the relationship between a central 
angle and an inscribed angle (the inscribed angle is 1/2 the central 
angle).  I would like to see that proven.  Where can I find a proof or 
do you know how to prove it or can you offer a hint as to what path to 
take to prove it?

Thanks,
Daniel

Date: 2/14/96 at 18:36:47
From: Doctor Byron
Subject: Re: Geometry Question

Hi Daniel,

Here's an idea for you to go on:  Draw a circle with center C.  Now 
add an arc AB to the circle.  Draw the radii AC and BC which connect the
endpoints of the arc to the center.  Now add the vertex of the inscribed
angle, I, and draw AI and BI.  Finally, add the diameter through C and I.

You will notice that you now have two congruent isoceles triangles to 
work with (since CB, AB, and CI are all radii, they are congruent).  
Using the nature of these triangles, and remembering the fact that the 
angles in all triangles sum to 180 degrees, you should be able to find 
your proof.

Good luck, and keep up the good work in math!

-Doctor Byron,  The Math Forum