Angle Properties of Circles

CIRCLES IT LESSON 1 - ANGLE PROPERTIES OF CIRCLES
(SEC 3 ELEMENTARY MATHEMATICS)

Software: JavaSketchpad (JSP)
Thinking Skills: Induction
At the end of the lesson, the students should be able to infer by induction 4 angle properties of circles:
(1) Angle at Centre,
(2) Angle in Semicircle,
(3) Angles in Same Segment,
(4) Angles in Opposite Segments.

A. ANGLE PROPERTY 1 (ANGLE AT CENTRE)

It takes a few seconds to initialise the JavaSketch below. But if the toolbar at the bottom reads, "Applet not found", click here to troubleshoot.

JavaSketchpad cannot support reflex angles. So you may choose to ignore reflex angles or to open the GSP file (IT3EMCirclesAngleCentre.gsp) using the Geometer's Sketchpad software which has a free evaluation version. Click on the hyperlink above and choose the option "Open it" instead of "Save it to disk". If you are using GSP for the first time, a dialogue box "Open With" will appear. If you don't know what to do, click here.

Sorry, this page requires a Java-compatible web browser.

1. The first JavaSketch (IT3EMCirclesAngleCentre.gsp) shows a circle with centre O, the angle at the centre, Angle(AOB), and the angle at the circumference, Angle(AXB).

2. Move the points A and B anywhere along the circumference of the circle (but not over the point X) to get different values for Angle(AOB) and Angle(AXB). Also Angle(AOB) should not exceed 180°.

Note: For angle at centre to be greater than 180°, open the GSP file IT3EMCirclesAngleCentre.gsp (see above). Move the point A or B to get the required angle. The GSP sketch also shows the angles being shaded which JavaSketchpad cannot support.

3. Tabulate the results in the Worksheet, giving your answers to one decimal place. You should choose at least one value of Angle(AOB) to be more than 180° if you have the GSP software to open the above GSP file.

Question 1: Infer by induction the relationship between the angle at the centre, Angle(AOB), and the angle at the circumference, Angle(AXB).

Click here for answer to Angle Property 1.

Question 2: Move the point X anywhere along the circumference of the circle (but not over A or B). What changes (if any) do you observe about the values of Angle(AOB) and Angle(AXB)? What conclusion can you draw from this?

Question 3: Move the point R towards and then away from the centre O in order to change the size of the circle. What changes (if any) do you observe about the values of Angle(AOB) and Angle(AXB)? What conclusion can you draw from this?

B. ANGLE PROPERTY 2 (ANGLE IN SEMICIRCLE)

1. Using the same JavaSketch above, move the point B until Angle(AOB) is 180°. The angle at the circumference, Angle(AXB), is now the angle in the semicircle since AB is the diameter.

Note: To get Angle(AOB) to be exactly 180°, you may have to make the line AOB horizontal or vertical.

2. Tabulate the values of Angle(AOB) and Angle(AXB) in the Worksheet.

Question 4: Move the point X anywhere along the circumference of the circle (but not over A or B). What changes (if any) do you observe about the value of Angle(AXB)? Infer by induction the angle in a semicircle.

Click here for answer to Angle Property 2.

C. ANGLE PROPERTY 3 (ANGLES IN SAME SEGMENT)

Sorry, this page requires a Java-compatible web browser.

1. The second JavaSketch (IT3EMCirclesAngleSegment.gsp) shows a circle with centre O and the angles in the same segment, Angle(AXB) and Angle(AYB).

2. Drag the points A, B, X and Y to observe. Or click on the animation buttons. To change the size of the circle, drag the point R.

Note: Do not move X or Y over A or B (so that the two angles remain in the same segment).

3. Tabulate the results in the Worksheet, giving your answers to one decimal place.

Question 5: Infer by induction the relationship between angles in the same segment.

Click here for answer to Angle Property 3.

D. ANGLE PROPERTY 4 (ANGLES IN OPPOSITE SEGMENTS)

1. Using the same JavaSketch above, move the point Y over the point B so that Angle(AXB) and Angle(AYB) are now in opposite segments.

2. Move A and B anywhere along the circumference of the circle (but not over X or Y) to get different values for Angle(AXB) and Angle(AYB). At the same time, you can also move the points X and Y along the circumference (but not over A or B) and the point R to change the size of the circle. Tabulate the results in the Worksheet, giving your answers to one decimal place.

Question 6: Infer by induction the relationship between angles in opposite segments.

Click here for answer to Angle Property 4.

[Return to top]

IT Menu | GSP | LiveMath

Question 1:	Infer by induction the relationship between the angle at the centre, Angle(AOB), and the angle at the circumference, Angle(AXB).
	Click here for answer to Angle Property 1.

Question 2:	Move the point X anywhere along the circumference of the circle (but not over A or B). What changes (if any) do you observe about the values of Angle(AOB) and Angle(AXB)? What conclusion can you draw from this?

Question 3:	Move the point R towards and then away from the centre O in order to change the size of the circle. What changes (if any) do you observe about the values of Angle(AOB) and Angle(AXB)? What conclusion can you draw from this?

1.	Using the same JavaSketch above, move the point B until Angle(AOB) is 180°. The angle at the circumference, Angle(AXB), is now the angle in the semicircle since AB is the diameter.
	Note: To get Angle(AOB) to be exactly 180°, you may have to make the line AOB horizontal or vertical.

2.	Tabulate the values of Angle(AOB) and Angle(AXB) in the Worksheet.

Question 4:	Move the point X anywhere along the circumference of the circle (but not over A or B). What changes (if any) do you observe about the value of Angle(AXB)? Infer by induction the angle in a semicircle.
	Click here for answer to Angle Property 2.

Question 5:	Infer by induction the relationship between angles in the same segment.
	Click here for answer to Angle Property 3.

1.	Using the same JavaSketch above, move the point Y over the point B so that Angle(AXB) and Angle(AYB) are now in opposite segments.

2.	Move A and B anywhere along the circumference of the circle (but not over X or Y) to get different values for Angle(AXB) and Angle(AYB). At the same time, you can also move the points X and Y along the circumference (but not over A or B) and the point R to change the size of the circle. Tabulate the results in the Worksheet, giving your answers to one decimal place.