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प्रश्न
Two circles intersect each other at points A and B. Their common tangent touches the circles at points P and Q as shown in the figure. Show that the angles PAQ and PBQ are supplementary.
उत्तर
Join AB.
PQ is the tangent and AB is a chord
∴ ∠QPA = ∠PBA ...(i) (Angles in alternate segment)
Similarly,
∠PQA = ∠QBA ...(ii)
Adding (i) and (ii)
∠QPA + ∠PQA = ∠PBA + ∠QBA
But, in ΔPAQ,
∠QPA + ∠PQA = 180° – ∠PAQ ...(iii)
And ∠PBA + ∠QBA = ∠PBQ ...(iv)
From (iii) and (iv)
∠PBQ = 180° – ∠PAQ
`=>` ∠PBQ + ∠PAQ = 180°
`=>` ∠PBQ + ∠PBQ = 180°
Hence ∠PAQ and ∠PBQ are supplementary.
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