Angles in Alternate Segments (Proof)

Case 1: Semicircle

Below shows a circle, with centre O.
The points A, B, C and X are points on the circumference of the circle.
The line PQ is the tangent to the circle at A.

You can:

• shift the point O to change the position of the circle
• shift the point X to change the size of the circle
• shift the point A to change the position of the semicircle ACB
• shift the point C to change its position

Instructions:

1. Adjust the position of the semicircle ACB.
2. Adjust the position of the point C.
3. Check all boxes.
4. Repeat steps 1 ~ 3 for other positions of the semicircle ACB and the point C.

Note: To see the animation, check all boxes, then click on the "play" button (at bottom left).



Download the GeoGebra file: http://www.geogebratube.org/material/download/format/file/id/41936


Case 2: Major Segment

Below shows a circle, with centre O.
The points A, B, C and X are points on the circumference of the circle.
The line PQ is the tangent to the circle at A.

You can:

• shift the point O to change the position of the circle
• shift the point X to change the size of the circle
• shift the points A and B to change the position and size of the segment ACB
• shift the point C to change its position

Instructions:

1. Adjust the position and size of the segment ACB.
2. Adjust the position of the point C.
3. Check all boxes.
4. Repeat steps 1 ~ 3 for other positions and sizes of the segment ACB and the point C.

Note: To see the animation, check all boxes, then click on the "play" button (at bottom left).



Download the GeoGebra file: http://www.geogebratube.org/material/download/format/file/id/41895


Case 3: Minor Segment

Below shows a circle, with centre O.
The points A, B, C and X are points on the circumference of the circle.
The line PQ is the tangent to the circle at A.

You can:

• shift the point O to change the position of the circle
• shift the point X to change the size of the circle
• shift the points A and B to change the position and size of the segment ACB
• shift the point C to change its position

Instructions:

1. Adjust the position and size of the segment ACB.
2. Adjust the position of the point C.
3. Check all boxes.
4. Repeat steps 1 ~ 3 for other positions and sizes of the segment ACB and the point C.

Note: To see the animation, check all boxes, then click on the "play" button (at bottom left).



Download the GeoGebra file: http://www.geogebratube.org/material/download/format/file/id/41896