Question

In the adjoining figure, two circles intersect each other at point M and N. Secants drawn from m and n intersect circles at point R,S,T and U .Prove seg PS ll seg QR.

Answer 1

from the center O of smaller circle, draw seg OT perpendicular to seg PS. join op ,os and seg mn
proof : quadrilateral SMNP is a cyclic quadrilateral.
angle MSP is equal to angle MNQ....... (corollary of cyclic quadrilateral theorem)
also, quadrilateral MNQR is a cyclic quadrilateral.
angle MNQ ➕ angle angle P is equal to 180.
.....Theorem of cyclic quadrilateral
angle MSP ➕ angle S is equal to 180
but they are a pair of interior angles on the same side of transversal SR on lines RQ and PS
therefore PS is parallel to QR

Answer 2

Answer:

quadrilateral SMNP is a cyclic quadrilateral.

angle MSP is equal to angle MNQ....... (corollary of cyclic quadrilateral theorem)

also, quadrilateral MNQR is a cyclic quadrilateral.

angle MNQ + angle angle P is equal to 180.

.....Theorem of cyclic quadrilateral

angle MSP + angle S is equal to 180

but they are a pair of interior angles on the same side of transversal SR on lines RQ and PS

therefore PS is parallel to QR