State and prove degree measure theorem

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State and prove degree measure theorem

Posted by Avni Rajput 2 years, 2 months ago

CBSE > Class 09 > Mathematics

3 answers

Prachi Bohra 2 years, 2 months ago

This is theorem 10.8 of circles

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Prachi Bohra 2 years, 2 months ago

Proof : There will be three cases as (i) PQ⌢is a minor arc (ii)PQ⌢ is a semi-circle (iii)PQ⌢ is a major arc [Image Here ] In each of these three cases, the exterior angle of a triangle is equal to the sum of two interior opposite angles. Therefore, ∠POM=∠PRO+∠RPO ............ (1) ∠MOQ=∠ORQ+∠RQO.......... (2) In △OPR and △OQR Now, OP = OR and OR = OQ (radii of the same circle) ∠PRO=∠RPO and ∠ORQ=∠RQO (angles opposite to the equal sides are equal) Hence, ∠POM=2∠PRO .......... (3) And ∠MOQ=2∠ORQ ............ (4) Case (1) : adding equations (3) and (4) we get ∠POM+∠MOQ=2∠PRO+2∠ORQ ∠POQ=2(∠PRO+∠ORQ)=2∠PRQ ∠POQ=2∠PRQ Hence Proved.

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Prachi Bohra 2 years, 2 months ago

Degree Measure Theorem states that angle subtended by an arc of a circle at the centre is twice the angle subtended by it at any point on the circle

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CBSE > Class 09 > Mathematics