Circle Geo qu (1 Viewer)

[Giant Lobster]

I cant do the following three, any help in any wud be appreciated!

1) PQ, RS and LM are three parallel chords in a circle. Prove that triangle PRL is congruent to triangle QSM

2) A circle has chords AD, AB, AC and DC. AC bisects angle BAD and cuts the circle at C. AB and DC are produced to meet at X. Given BX = AD prove triangle ACD is congruent to triangle BCX. (me fink this one impossible / book misprint; prove me wrong )

3) AB and CB are two parallel chords of a circle. CA and DB meet, when produced at X. AD and BC intersect at Y. Prove XY (produced if necessary) will pass through the centre O of the circle.

thanks in advance.

 

[ND]

1.) PRS + RPQ = 180 (cointerior)
PRS + PQS = 180 (interior angles of cyclic quad are supp)
.'. RPQ = PQS

Similarly MSR = LRS
.'. LRP = MSQ (cos they are the sums of the above angles)

Also PQRS, LRMS are trapeziums (2 sets of angles are equal) and so RP = QS and RL = SM.
.'. triangle PRL is congruent to triangle QSM (2 sides and included angles are equal).

2.) Yep this is incorrect:

Let a be DAC, now XCB = 2a (ext angle = opp int angle)
and let b be ADC, so CBX = b (")
now in triangle DAC, two angles are a and b, but triangle BCX has angles 2a and b, which obviously isn't right.

3. If AB and AC are parallel, then AB=AC.

 

[Giant Lobster]

thanks for ur help, u gave me a huge hint

but im not sure about this:

Also PQRS, LRMS are trapeziums (2 sets of angles are equal) and so RP = QS and RL = SM.

is that some kind of theorem or somefin?