Question

In figure, O is the centre of the circle such that $\angle AOC=130^\circ$, then $\angle ABC=\dots\dots$

• A. $130^\circ$
• B. $115^\circ$
• C. $65^\circ$
• D. $165^\circ$

Hint:

Always be careful to distinguish between the angle subtended by the minor arc and the major (reflex) arc at the center when applying the theorem. The angle at the circumference corresponds to the arc it subtends.

Answer 1

The Correct Option is B

Step 1: Understand the relationship between the angle at the center and the angle at the circumference.
The angle subtended by an arc at the center of a circle is double the angle subtended by it at any point on the remaining part of the circle. Step 2: Calculate the reflex angle $\angle AOC$.
The given $\angle AOC$ is $130^\circ$. This is the angle subtended by the minor arc AC at the center.
The reflex angle $\angle AOC$ (the angle formed by the major arc AC at the center) is $360^\circ - 130^\circ = 230^\circ$. Step 3: Apply the theorem for the angle subtended by the major arc.
The angle subtended by the major arc AC at the center is the reflex $\angle AOC = 230^\circ$.
The angle subtended by the major arc AC at point B on the circumference is $\angle ABC$.
According to the theorem, $\text{reflex } \angle AOC = 2 \times \angle ABC$. Step 4: Solve for $\angle ABC$.
$230^\circ = 2 \times \angle ABC$
$\angle ABC = \frac{230^\circ}{2}$
$\angle ABC = 115^\circ$
Step 5: Compare the result with the given options.
The calculated angle $\angle ABC$ is $115^\circ$, which matches option (2). (2) 115\textdegree