Question

In the above figure, \( \triangle ABC \) is inscribed in arc \( ABC \). If \( \angle ABC = 60^\circ \), find \( m\angle AOC \): 

Hint:

For inscribed angles, remember that \( \text{Inscribed Angle} = \frac{1}{2} \times \text{Intercepted Arc} \). For central angles, \( \text{Central Angle} = \text{Intercepted Arc} \).

Answer 1

Step 1: Using the property of inscribed angles, the measure of an inscribed angle is half the measure of the arc it subtends: \[ \angle ABC = \frac{1}{2} m(\text{arc AXC}). \] Substituting \( \angle ABC = 60^\circ \): \[ 60^\circ = \frac{1}{2} m(\text{arc AXC}). \] Step 2: Solving for \( m(\text{arc AXC}) \): \[ m(\text{arc AXC}) = 2 \times 60^\circ = 120^\circ. \] Step 3: Using the property of central angles, the measure of a central angle is equal to the measure of the arc it subtends: \[ m\angle AOC = m(\text{arc AXC}) = 120^\circ. \] Hence, \( m\angle AOC = 120^\circ \).