In the adjoining figure circles with centres X and Y touch each other at point Z. A secant passing through Z intersects the circles at points A and B respectively. Prove that, radius XA || radius YB. Fill in the blanks and complete the proof.

Construction: Draw segments XZ and ..YZ........ .

Proof: By theorem of touching circles, points X, Z, Y are ..concyclic........ .

..∠ YZB ........vertically opposite angles

Let ..... (I)

Now, seg XA ≅seg XZ ........ (...radius of the same circle.......)

= ....∠ XZA...... = a ........ (isosceles triangle theorem) (II)

similarly, seg YB ≅ .YZ......... ........ (.radius of the same circle.........)

.∠ZBY......... = a ........ (.isosceles triangle theorem.........) (III)

∴from (I), (II), (III),

∠ XAZ = .∠ ZBY.........

∴radius XA || radius YB .......... (..since alternate interior angles are equal........)