Second circle theorem: 'Angle in a semicircle'

Angle in a semicircle

Second circle theorem:

There are 2 Geogebra windows here; newer one just below, & original lower down.

For the first one, try moving the point P about. Where does P have to be for angle α to be acute? and where must P be for angle α to be obtuse? So where must P be to make angle α 90 degrees?

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

For the window below, what's the size of the angle α ('angle in a semicircle')? You can use the cursor to move point D around the circumference;

It doesn't change angle α though!!
It is always 90 degrees.
This theorem is a special case of theorem 1 - 'angle at the centre is twice the angle at the circumference', because the centre angle is 180 degrees.

If you need them, the theorems are summarised here

Tim Devereux, Created with GeoGebra