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.Then, the angles PAQ and PBQ are supplementary.

• A. True
• B. False

Answer 1

The correct option is A

True

Given - Two circles intersect each other at A and B A common tangent touches the circles at P and Q. PA, PB, QA and QB are joined.

Construction - Join AB,

Proof - PQ is the tangent and AB is the chord

$\therefore \angle QPA=\angle PBA$ (alternate segment) ....(i)

similarly we can prove that

$\angle PQA=\angle QBA\dots \left(ii\right)$

Adding (i ) and (ii), we get

$\angle QPA+\angle PQA=\angle PBA+\angle QBA$

But $\angle QPA+\angle PQA={180}^{\circ }-\angle PAQ=\dots \left(iii\right)$

(In $\Delta PAQ$)

and $\angle PBA+\angle QBA=\angle PBQ\dots \left(iv\right)$

from (iii) and (iv)

$\angle PBQ={180}^{\circ }-\angle PAQ$

$\Rightarrow \angle PBQ+\angle PAQ={180}^{\circ }$

$\Rightarrow \angle PAQ+\angle PBQ={180}^{\circ }$

Hence $\angle PAQ$ and $\angle PBQ$ are supplementary.