2.2 Angle Properties of a Circle

• Arc, chord and segment
• Angle subtended by the same arc
• Angle subtended by the same chord
• Angles in a semicircle
• Cyclic quadrilateral

• identify an arc, chord, and segment in a circle
• relate angles subtended by the same arc at the circumference
• relate the angle subtended by an arc at the centre and at the circumference
• determine angles subtended by a chord and a diameter at the circumference
• determine angles in a cyclic quadrilateral
• apply the angle properties of a circle in different situations
• reflect on the use of angle properties of a circle in different situations

“How is the knowledge of angle properties of a circle used in day to day life?”

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1. Arc, chord and segment
   40 min · 1 practice tasks · quiz
   Name an arc, a chord, a segment and a sector, and separate the minor part of a circle from the major part.
2. The angle an arc subtends
   40 min · 1 practice tasks · quiz
   Say which arc an angle stands on, at the centre or at a point on the remaining circumference.
3. The centre angle is twice
   40 min · 1 practice tasks · quiz
   The angle at the centre is twice the angle at the circumference when both stand on the same arc.
4. The angle on the other arc
   40 min · 1 practice tasks · quiz
   Half the reflex angle at the centre gives the circumference angle that stands on the major arc.
5. Angles in the same segment
   40 min · 1 practice tasks · quiz
   Two angles in the same segment, standing on the same arc, are equal.
6. Both segments of one chord
   40 min · 1 practice tasks · quiz
   Angles in opposite segments of the same chord add to 180 degrees.
7. The angle in a semicircle
   40 min · 1 practice tasks · quiz
   A diameter subtends 90 degrees at every point on the remaining circumference, and a right angle at the circumference means the opposite side is a diameter.
8. Equal radii, equal base angles
   40 min · 1 practice tasks · quiz
   Use equal radii to make isosceles triangles, then combine them with the right angle in a semicircle.
9. Opposite angles of a cyclic quadrilateral
   40 min · 1 practice tasks · quiz
   If all four vertices lie on a circle, each pair of opposite angles adds to 180 degrees.
10. The exterior angle
    40 min · 1 practice tasks · quiz
    The exterior angle of a cyclic quadrilateral equals the opposite interior angle.
11. Choose the circle theorem
    40 min · 1 practice tasks · quiz
    In one figure, halve the centre angle, copy an angle in the same segment, and subtract from 180 for the other segment.
12. A window, a wheel and a seat
    40 min · 1 practice tasks · quiz
    Use a diameter to fix a right angle in a circular window, a hub angle to fix a rim angle, and a circle to fix the opposite corner of a four-sided seat.