Question

In the figure \(∠BAO 30°, ∠BCO=40°\) then \(∠AOC = ?\)

• A. \(100°\)
• B. \(120°\)
• C. \(140°\)
• D. \(150°\)

Answer 1

The Correct Option is C

To solve the problem, we need to find the measure of \( \angle AOC \) given that \( \angle BAO = 30^\circ \) and \( \angle BCO = 40^\circ \). The figure shows a circle with center \( O \), and points \( A \), \( B \), and \( C \) on the circumference.

Step 1: Understand the Geometry
\( O \) is the center of the circle.
\( \angle BAO \) and \( \angle BCO \) are angles formed by radii and chords.
We need to find \( \angle AOC \), which is the central angle subtended by the arc \( AC \).
Step 2: Use the Inscribed Angle Theorem The key property here is that the central angle is twice the inscribed angle subtended by the same arc. However, we first need to find the measure of the inscribed angle \( \angle ABC \).
Step 3: Find \( \angle ABC \) In \( \triangle AOB \):
\( OA = OB \) (radii of the circle).
Therefore, \( \triangle AOB \) is isosceles, and the base angles are equal.
Given \( \angle BAO = 30^\circ \), the other base angle \( \angle ABO \) is also \( 30^\circ \).
The sum of angles in a triangle is \( 180^\circ \): \[ \angle AOB = 180^\circ - \angle BAO - \angle ABO = 180^\circ - 30^\circ - 30^\circ = 120^\circ \] Similarly, in \( \triangle COB \):
\( OC = OB \) (radii of the circle).
Therefore, \( \triangle COB \) is isosceles, and the base angles are equal.
Given \( \angle BCO = 40^\circ \), the other base angle \( \angle CBO \) is also \( 40^\circ \).
The sum of angles in a triangle is \( 180^\circ \): \[ \angle COB = 180^\circ - \angle BCO - \angle CBO = 180^\circ - 40^\circ - 40^\circ = 100^\circ \]
Step 4: Find \( \angle AOC \) The central angle \( \angle AOC \) is the sum of \( \angle AOB \) and \( \angle COB \): \[ \angle AOC = \angle AOB + \angle COB = 120^\circ + 100^\circ = 220^\circ \] However, since \( \angle AOC \) is the reflex angle, the actual central angle we are interested in is: \[ \angle AOC = 360^\circ - 220^\circ = 140^\circ \]

Final Answer: \[ {140^\circ} \]