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 |  ​​ |
AB is chord of a circle with centre O and C and D lie on the same side of AB.If | |||
| Right Option : C | |||
| View Explanation | |||
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| Right Option : A | |||||
| View Explanation | |||||
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| Right Option : D | |||||
| View Explanation | |||||
Students / Parents Reviews [10]
About Abhyas metholodology the teachers are very nice and hardworking toward students.The Centre Head Mrs Anu Sethi is also a brilliant teacher.Abhyas has taught me how to overcome problems and has always taken my doubts and suppoeted me.
Shreya Shrivastava
8thI have spent a wonderful time in Abhyas academy. It has made my reasoning more apt, English more stronger and Maths an interesting subject for me. It has given me a habbit of self studying
Yatharthi Sharma
10thMy experience was very good with Abhyas academy. I am studying here from 6th class and I am satisfied by its results in my life. I improved a lot here ahead of school syllabus.
Ayan Ghosh
8thIt was good as the experience because as we had come here we had been improved in a such envirnment created here.Extra is taught which is beneficial for future.
Eshan Arora
8thA marvelous experience with Abhyas. I am glad to share that my ward has achieved more than enough at the Ambala ABHYAS centre. Years have passed on and more and more he has gained. May the centre flourish and develop day by day by the grace of God.
Archit Segal
7thMy experience with Abhyas is very good. I have learnt many things here like vedic maths and reasoning also. Teachers here first take our doubts and then there are assignments to verify our weak points.
Shivam Rana
7thIt was a good experience with Abhyas Academy. I even faced problems in starting but slowly and steadily overcomed. Especially reasoning classes helped me a lot.
Cheshta
10thOne of the best institutes to develope a child interest in studies.Provides SST and English knowledge also unlike other institutes. Teachers are co operative and friendly online tests andPPT develope practical knowledge also.
Aman Kumar Shrivastava
10thAbhyas Methodology is very good. It is based on according to student and each child manages accordingly to its properly. Methodology has improved the abilities of students to shine them in future.
Manish Kumar
10thBeing a parent, I saw my daughter improvement in her studies by seeing a good result in all day to day compititive exam TMO, NSO, IEO etc and as well as studies. I have got a fruitful result from my daughter.
