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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. - ICSE Class 10 - Mathematics

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ConceptTangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers

Question

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.

Solution

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 Δ PAQ,
`∠`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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Solution 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. Concept: Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.
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