1. AB and CD are two chords of a circle. When BA and DC are extended, they intersect at point P. Prove that \(\angle\)PCB = \(\angle\)PAD.

2. In the adjacent figure, AB and CD are two chords of a circle. When BA and DC are extended, they intersect at point P. Prove that \(\angle\)PCB = \(\angle\)PAD.

3. AOB is the diameter of a circle. AC and BD are two chords that intersect at point E. If \(\angle\)COD = 40°, find the value of \(\angle\)CED.

4. AB and CD are two chords of a circle with center O. When extended, they intersect at point P. Prove that \(\angle AOC - \angle BOD = 2\angle BPC\).

5. "Chords PQ and RS of a circle intersect each other at point X inside the circle. By joining P to S and R to Q, prove that triangles ∆PXS and ∆RSQ are similar. From this, prove that: PX × XQ = RX × XS Or, when two chords of a circle intersect internally, the product of the two segments of one chord is equal to the product of the two segments of the other chord.

6. Two chords AB and CD of a circle with center O intersect each other at point P. Prove that \(\angle AOD + \angle BOC = 2\angle BPC\) If \(\angle AOD\) and \(\angle BOC\) are supplementary, then prove that the two chords are perpendicular to each other.

7. When the chords AB and CD of a circle with center O are extended, they intersect each other at point P. Prove that \(\angle\)AOC − \(\angle\)BOD = 2\(\angle\)BPC.