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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

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)

∠ 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

Is there an error in this question or solution?

#### APPEARS IN

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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