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"Related Properties of Circles" Circle PPT Courseware 4
Question: What is the relationship between the degrees of the circumferential angles and the corresponding central angles?
(1) When the center of the circle is on one side of the circumference angle, prove: (the center of the circle is on the circumference angle)
Conclusion: The circumferential angle subtended by an arc is equal to half of the central angle subtended by it.
2. When the center of the circle is outside the circumferential angle
Tip: Can it be converted into the case of 1?
Draw the diameter BD through point B. From 1 we can get:
∠ABD =1/2∠AOD, ∠CBD =1/2∠COD,
∴ ∠ABC =1/2∠AOC.
Circle Angle Theorem
The circumferential angle subtended by an arc is equal to half the central angle subtended by it.
Circular angles subtended by the same arc are equal
1. How many degrees is the circumferential angle subtended by a semicircle or diameter?
2.Is the chord subtended by the circumferential angle of 90° the diameter?
inference:
The circumferential angles subtended by a semicircle or a diameter are all equal, equal to 90° (right angle). The reverse is also true, that is, the chord subtended by a 90° circumferential angle is the diameter of the circle.
The concept of circular angle:
The angle where the vertex is on the circle and the two sides intersect the circle is called the circumferential angle.
Circle angle theorem:
In congruent circles or congruent circles, the circumferential angles subtended by congruent arcs or congruent arcs are equal to half of the central angle subtended by this arc.
inference:
The circumferential angles subtended by a semicircle or diameter are all equal, equal to 90°; the chord subtended by a 90° circumferential angle is the diameter of the circle.
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