Let us take a circle and a quadrilateral ABCD in it. In the quadrilateral, let us try to expand the angle at A by shifting B to B' and D to D'. What is the consequence at the angle C which is angle B' C D' now? More we expand the angle at A, the angle at C shrinks more. Similarly expansion of the angle at C results in contraction of angle at A, keeping the vertices of the quadrilateral on the given circle.
In this increase/decrease situations at A / C, will there be not the situation, when the angles at A and C are equal? What is that situation? Is not that an interesting situation? Let us now restart from this special situation. Make a measured increase in the angle at A. What is the measure of decrease in angle at C? Repeat the process with a new measured increases at A, make your observations. What do you discover? Write down your discovery, in the format of a theorem.
Using the theorem, the angle at A determines the angle at C. In a new activity proposed, let us not change the angle at A and so B and D don't move from their positions. Choose another point C' on the circle. By our theorem the angle at A determines the angle at C' also. What is the relation between the angles at C and C'? If we choose another point C'' on the arc BD, what do you observe about the angles in the same segment of a circle? Write down your discovery.
We discover that angles in same segment of a circle are all equal. So angles at C, C', C'', ... are all equal. Even at the point E, the point on the circle obtained by extending BO , the angle remains the same.
However, EO = AO. So \angle OEA = \angle AEO.
Thus angle at the centre, i.e., \angle AOB, which is the sum of \angle OEA and \angle AEO is actually twice the angle \angle AEO. Does it say something about the relation between the angle formed at the centre by a segment and angles at any point on the circle made by the same segment?
In this increase/decrease situations at A / C, will there be not the situation, when the angles at A and C are equal? What is that situation? Is not that an interesting situation? Let us now restart from this special situation. Make a measured increase in the angle at A. What is the measure of decrease in angle at C? Repeat the process with a new measured increases at A, make your observations. What do you discover? Write down your discovery, in the format of a theorem.
Using the theorem, the angle at A determines the angle at C. In a new activity proposed, let us not change the angle at A and so B and D don't move from their positions. Choose another point C' on the circle. By our theorem the angle at A determines the angle at C' also. What is the relation between the angles at C and C'? If we choose another point C'' on the arc BD, what do you observe about the angles in the same segment of a circle? Write down your discovery.
We discover that angles in same segment of a circle are all equal. So angles at C, C', C'', ... are all equal. Even at the point E, the point on the circle obtained by extending BO , the angle remains the same.
However, EO = AO. So \angle OEA = \angle AEO.
Thus angle at the centre, i.e., \angle AOB, which is the sum of \angle OEA and \angle AEO is actually twice the angle \angle AEO. Does it say something about the relation between the angle formed at the centre by a segment and angles at any point on the circle made by the same segment?
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