In the figure 3.87 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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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. Figure: The circles touch externally at Z; A is on the circle with centre X, B is on the circle with centre Y, and A–Z–B. Construction: Draw segments XZ and ………. . Proof: By theorem of touching circles, points X, Z, Y are ………. . therefore ∠ XZA cong ………. opposite angles Let ∠ XZA = ∠ BZY = a ….. (I) Now, seg XA ≅ seg XZ ……… (……….) therefore ∠ XAZ = ………. = a …….. (isosceles triangle theorem) (II) similarly, seg YB ≅ ………. …….. (……….) therefore ∠ BZY = ………. = a …….. (……….) (III) therefore from (I), (II), (III), ∠ XAZ = ………. therefore radius XA ∥ radius YB ………. (……….)

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Answer: ∠ XAZ = ∠ YBZ (alternate angles), so radius XA ∥ radius YB. Construction: Draw segments XZ and YZ. By the theorem of touching circles, points X, Z, Y are collinear. therefore ∠ XZA cong ∠ BZY (vertically opposite angles). Let ∠ XZA = ∠ BZY = a …… (I) Seg XA ≅ seg XZ (radii of the same circle) therefore ∠ XAZ = ∠ XZA = a (isosceles triangle theorem) …… (II) Similarly, seg YB ≅ seg YZ (radii of the same circle) therefore ∠ BZY = ∠ ZBY = a (isosceles triangle theorem) …… (III) From (I), (II), (III), ∠ XAZ = ∠ YBZ . These are alternate angles made by transversal AB, so radius XA ∥ radius YB (alternate angles test).

In the figure 3.87 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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