1. The measure of the central angle is twice the measure of an inscribed angle, subtended by the same arc.

Inscribed/central angle and minor/major arc animations:

• All 3 points that define an inscribed angle are on the circle.
• The vertex is on the centre of the circle and the other 2 points that define a central angle are on the circle.
• The inscritbe angles in the slideshow is <BAC and the central angle is <BOC . The endpoints BC are connected wiith arc BC. The inscribed angles share (are subtended by) arc BC.
• The interactive on the left allows you to change the vertex position for <BAC. The size of angles BAC and BOC will not change when you move vertex A. Compare the size of the two angles. Change the postion of B and C and compare the size of the two angles. If you can remember what you see, you will likely remember: The measure of the central angle is twice the measure of an inscribed angle, subtended by the same arc.

arc BC subtends central <BOC and inscribed <BAC. Therefore the measure of <O is twice the measure of <B.

Interactive Activity

• Adjust the size of the inscribed angles by dragging the points A, B, and C.
• <O will always be twice as big as <A.
• Notice the red arc that connects the pair of angles subtended by that arc.
• Hold the SHIFT key when you drag on the circumference of the circle to change the size of the circle.

Linked Source - Ron Blond

Download the newest java version if the applet fails to launch.

2. Any angle inscribed in a semicircle is always a right angle.

Sample Graphic:

arc BC is a semicircle. Therefore inscribed angle A is a right angle.

The central angle is 180^o. The inscribed angle is 90^o.

Remember: The measure of the central angle is twice the measure of an inscribed angle, subtended by the same arc.

2(90^o) = 180^o

or

180^o/2 = 90^o

Interactive Activity

• Move point A or C to make <O equal 180^o. <B will be 90^o.
  Adjust the size of the inscribed angles by dragging the points A, B and C.
• Hold the SHIFT key when you drag on the circumference of the circle to change the size of the circle.

Linked Source - Ron Blond