In the given figure, $AB$ and $CD$ are tangents to a circle centred at $O$. Is $\angle BAC = \angle DCA$? Justify your answer.

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Question

In the given figure, $AB$ and $CD$ are tangents to a circle centred at $O$. Is $\angle BAC = \angle DCA$? Justify your answer.

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Accepted Answer

Step 1: Circle with radii OA and OC drawn, showing equal angles
Step 2: Join $OA$ and $OC$. $OA = OC$ (radii), so $\angle OAC = \angle OCA$
Step 3: Also, $\angle OAB = \angle OCD = 90°$ (radius $\perp$ tangent)
Step 4: $\Rightarrow \angle OAC + \angle OAB = \angle OCA + \angle OCD \Rightarrow \angle BAC = \angle DCA$

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