Question

STATEMENT - 1 : The opposite side of a quadrilateral circumscribing a circle subtend supplementary angle at the centre of the circle.
STATEMENT - 2 : The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

• A. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1
• B. Statement - 1 is True, Statement - 2 is True : Statement 2 is NOT a correct explanation for Statement - 1
• C. Statement - 1 is True, Statement - 2 is False
• D. Statement - 1 is False, Statement - 2 is True

Answer 1

The correct option is B Statement - 1 is True, Statement - 2 is True : Statement 2 is NOT a correct explanation for Statement - 1

Statement 1 is correct:

Join the vertices of the quadrilateral $ABCD$ to the center of the circle.

In $△OAP$ and $△OAS$,

$AP=AS$ (Tangents from the same point)

$OP=OS$ (Radii of the circle)

$OA=OA$ (Common side)

$△OAP=△OAS$ (SSS congruence condition)

$\therefore \angle POA=\angle AOS$

$\angle 1=\angle 8$

Similarly we get,

$\angle 2=\angle 3$

$\angle 4=\angle 5$

$\angle 6=\angle 7$

Adding all these angles,

$\angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6+\angle 7+\angle 8={360}^{\circ }$

$⟹(\angle 1+\angle 8)+(\angle 2+\angle 3)+(\angle 4+\angle 5)+(\angle 6+\angle 7)={360}^{\circ }$

$⟹2\angle 1+2\angle 2+2\angle 5+2\angle 6={360}^{\circ }$

$⟹2(\angle 1+\angle 2)+2(\angle 5+\angle 6)={360}^{\circ }$

$⟹(\angle 1+\angle 2)+(\angle 5+\angle 6)={180}^{\circ }$

$⟹\angle AOB+\angle COD={180}^{\circ }$

Similarly, we can prove that $\angle BOC+\angle DOA={180}^{\circ }$

Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Statement 2 is correct as in quadrilateral $DSRO$, $\angle S=\angle R={90}^{\circ }$ and as $DS=DR$, $OS=OR$, $\angle SDO=\angle RDO$ and $\angle SOD=\angle ROD$ and also $\angle SDR+\angle SOR={180}^{\circ }$

Hence, statement 2 is also correct

But, it is not the correct explanation of statement 1 as the supplementary nature of tangent's angles does not relate to the angle subtended by the opposite sides at the center.