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.
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Question 1: | Infer by induction
the relationship between the angle at the centre, Angle(AOB), and the angle
at the circumference, Angle(AXB).
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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?
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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?
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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.
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Click here for answer to Angle Property 2. |
C. ANGLE PROPERTY 3 (ANGLES IN SAME
SEGMENT)
Question 5: | Infer by induction
the relationship between angles in the same segment.
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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. |
Question 6: | Infer by induction
the relationship between angles in opposite segments.
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