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.

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Solution

Construction: Draw segments XZ and YZ.

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

∴ ∠XZA ≅ ∠BZY (opposite angles)

Let ∠XZA = ∠BZY = a .....(I)

Now, seg XA ≅ seg XZ ..... (Radii of circle with centre X)

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

Similarly, seg YB ≅ seg YZ ..... (Radii of circle with centre Y)

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

∴ from (I), (II), (III),

∠XAZ = ∠ZBY

∴ radius XA || radius YZ ..... (Alternate angle test)

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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.

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.