Question

Is it right that the central angle of an arc will be the double of the angle made by the alternate arc?

Answer 1

Answer:

Yes...

Step-by-step explanation:

Theorem: Central Angle Theorem

The Central Angle Theorem states that the central angle from two chosen points A and B on the circle is always twice the inscribed angle from those two points. The inscribed angle can be defined by any point along the outer arc AB and the two points A and B.

Hope it helps u mate

Answer 2

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Given :

An arc PQ of a circle subtending angles POQ at the centre O and PAQ at a point A on the remaining part of the circle.

To prove : ∠POQ=2∠PAQ

To prove this theorem we consider the arc AB in three different situations, minor arc AB, major arc AB and semi-circle AB.

Construction :

Join the line AO extended to B.

Proof :

∠BOQ=∠OAQ+∠AQO       .....(1)

Also, in △ OAQ,

OA=OQ                  [Radii of a circle]

Therefore,

∠OAQ=∠OQA      [Angles opposite to equal sides are equal]

∠BOQ=2∠OAQ              .......(2)

Similarly, BOP=2∠OAP             ........(3)

Adding 2 & 3, we get,

∠BOP+∠BOQ=2(∠OAP+∠OAQ)

∠POQ=2∠PAQ               .......(4)

For the case 3, where PQ is the major arc, equation 4 is replaced by

Reflex angle, ∠POQ=2∠PAQ