The theorem says that "The measure of an exterior angle of a triangle is sum of the interior opposite angles". Select the correct option from the followings

∠ POQ = 2 ∠ PAQ

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∠ POQ = 2 ∠ PAB

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∠ POQ = 2 ∠ QAB

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∠ POQ = ∠ PAQ

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Solution

The correct option is A $∠ POQ = 2 ∠ PAQ$
According to the theorem,
$∠ POB = ∠ PAO + ∠ APO$
Similarily,
$∠ QOB = ∠ QAO + ∠ AQO$
Now,
$\Rightarrow ∠ PAO = ∠ APO$ ,
$∴ ∠ POB = 2 ∠ PAO$
$\Rightarrow ∠ QAO = ∠ AQO$ ,
$∴ ∠ QOB = 2 ∠ QAO$
(as same sides have similar angles)
And
$∠ POQ = ∠ POB + ∠ QOB$
$∠ PAQ = ∠ PAO + ∠ QAO$
So,
$∠ POQ = ∠ POB + ∠ QOB$
$= 2 ∠ PAO + 2 ∠ QAO$
$= 2 (∠ PAO + ∠ QAO)$
$= 2 ∠ PAQ$

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