Assertion (A) TA and TB are two tangents drawn from an external point T to a circle with centre 'O'. If $\angle TBA = 75^\circ$ then $\angle ABO = 25^\circ$. Reason (R) The tangent drawn at any point of a circle is perpendicular to the radius through the point of contact.

Step 1: Analyze the given information

We are given that TA and TB are tangents drawn from an external point T to a circle with center O. We are also given that the angle $\angle TBA = 75^\circ$. We need to verify if the assertion that $\angle ABO = 25^\circ$ is true.

Step 2: Apply properties of tangents and radii

According to the reason (R), the tangent drawn at any point of a circle is perpendicular to the radius through the point of contact. Therefore, the radius OB is perpendicular to the tangent TB at the point of contact B, which means $\angle OBT = 90^\circ$.

Step 3: Calculate $\angle$ ABO

We know that $\angle OBT$ is the sum of $\angle OBA$ and $\angle TBA$. Substituting the known values, we get $90^\circ = \angle OBA + 75^\circ$. Solving for $\angle OBA$, we find that $\angle OBA = 15^\circ$.

Step 4: Evaluate the assertion

Our calculation shows that $\angle ABO = 15^\circ$. The assertion states that $\angle ABO = 25^\circ$. Since $15^\circ \neq 25^\circ$, the assertion (A) is false.

Step 5: Evaluate the reason

The reason (R) states that the tangent drawn at any point of a circle is perpendicular to the radius through the point of contact. This is a fundamental theorem in geometry and is a true statement.

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