In the given figure, AB and CD are tangents to a circle centred at O. Is $\angle BAC = \angle DCA$? Justify your answer.
Generated by claude-sonnet-4-6 · 2026-06-15 10:22 · grounding rag
Model Answer
Yes, ∠BAC = ∠DCA.
Justification: Since AB and CD are tangents to the circle, the two tangent lines form a pair of parallel tangents (both touching the circle from opposite sides). AC is a transversal cutting these parallel lines.
Therefore, ∠BAC and ∠DCA are alternate interior angles, which makes them equal.
$$\angle BAC = \angle DCA$$
Source: Chapter 10, Tangents to a Circle
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Explanation
- The key observation is that AB ∥ CD (two tangents on opposite sides of a circle are parallel, as the radius to each point of contact is perpendicular to the tangent, making both tangents perpendicular to the same line through O).
- AC acts as a transversal; alternate interior angles formed by a transversal with parallel lines are equal.
- Examiners want you to state that AB ∥ CD and then invoke the alternate interior angles property — both steps are needed for full marks.