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........)

Question

Philoid · Accepted Answer

Construction: Draw segments XZ and YZ.

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

_∠ _{XZA =} ∠ YZB {vertically opposite angles}

Let ∠ XZA = ∠ BZY = a (I)

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

_∵ _∠ _XAZ = ∠ XZA = a (isosceles triangle theorem) (II)

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

∴ ∠ BZY = ∠ZBY = a (isosceles triangle theorem) (III)

∴from (I), (II), (III),

∠ XAZ = ∠ ZBY

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