Angles in the Same Segment are Equal
Angles in the Same Segment are Equal
Theorem 1 Angles in the same segment of a circle are equal.
Given A circle C(O,r), an arc PQ and two angles To prove Construction Join OP and OQ. PROOF We know that the angle subtended by an arc at the centre is double the angle subtended by the arc at any point in the remaining part of the circle . So, in figure, we have
Hence proved. |  |
| Converse Theorem: If a line segmentjoining two points subtends equal angles at two other points lying on the same side of the line containing the line segment. the four points lie on the circle. | |
Given: A line segment AB and two points C and D such that To Prove: The points A, B, C and D lie on a circle or the points are concyclic Construction: Draw a circle Which passes through A, B C Proof As the circle passes through A, B and C, let us assume that it does not pass through D. Then either the circle will intersect AD at F or will cut the line segment AD extended to E Case I: THe circle intersects AD at F . Then A,C,F,B lie on the circle
but From Equation 1 and 2 we get This is not possible because Therfore D and F should coincide Case 2: The circle cuts the line segment AD extended to E Then A,C,E,B lie on the circle
but From Equation 3 and 4 we get This is not possible because Therfore D and E should coincide Hence the circle has to pass through A, B, C and C or These points are concyclic |  |
Illustration: If in the figure Solution: In the figure we have a circle with center O and AC is the arc The arc AC subtends angle at point B and D on the circle Now Therefore As |  ​​ |
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| Right Option : D | |||||
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| Right Option : B | |||||
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| Right Option : C | |||||
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