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Secondary 3 (Grade 9) Geometry
Guiding Question
Angles, Angles in the Circle, What Do You Know About Them?
Topic
Angle Properties of Circles
Software Required
Cabri-Jr. Application for the TI-83 Plus Graphics Calculator
Level
Secondary 3 (Grade 9)
Duration
Three 40-minute periods
Objective
To investigate the following angle properties of circles:
1. Angles in the same segment are equal.
2. Angle at the center is twice angle at the circumference.
3. Opposite angles of a cyclic quadrilateral are supplementary.
Pre-requisite Knowledge
Students have been introduced to the following terms in Geometry:
1. Arc – major and minor
2. Chord and diameter
3. Segment – major and minor
4. Sector – major and minor
5. Angle at the center
6. Angle at the circumference
7. Cyclic quadrilateral
Construction
Property 2:
1. Draw a circle.
2. Construct a segment from the center of the circle, O, to a point on the
circumference (A).
3. Construct another segment from the center of the circle to another point
on the circumference (B).
4. Construct two segments to join A and B to a new point on the
circumference (C).
5. Measure the angle ACB.
6. Measure also the angle AOB.
7. Observe the relationship between the two angles.
8. Animate point A or point B.
NIE2004/Inquiry Learning/TCHS/Sec3(Grade9)/Mok SL
9. Stop the animation at different points on the circumference.
10. Observe the relationship between the two angles at the different points.
11. Discussion: What conjecture can you make about the relationship
between the two angles?
Property 1:
1. How can you use the construction in Property 2 to prove Property 1?
2. Discussion: In groups of two or three, use the software to make the
connection.
Property 3:
1. Draw a circle.
2. Draw a quadrilateral within the circle with its vertices on the circumference
of the circle drawn earlier.
3. Measure two opposite angles within the quadrilateral.
4. Animate the vertex of either one of the two angles.
5. Note the values of the measurements of the two opposite angles as the
animation progresses.
6. Discussion: What conjecture can you make about the relationship of the
two opposite angles? As you animate the vertex of one of the two angles,
why does the value of the angle toggle between two values only? Hint:
Use the properties earlier to explain.
NIE2004/Inquiry Learning/TCHS/Sec3(Grade9)/Mok SL