Question

In the figure, \(O\) is the centre of the circle and \(AC\) is the diameter. The line \(FEG\) is tangent to the circle at \(E\). If \(\angle GEC = 52^\circ\), find the value of \(\angle E + \angle C\).

• A. \(154^\circ\)
• B. \(156^\circ\)
• C. \(166^\circ\)
• D. \(180^\circ\)

Hint:

Remember that angle between tangent and chord equals angle in alternate segment. Use triangle angle sum in circle-based triangle.

Answer 1

The Correct Option is B

Step 1: Use geometry of the circle
We are given: - \(AC\) is the diameter of the circle, hence triangle \(AEC\) is a right triangle (angle in a semicircle). \[ \angle AEC = 90^\circ \] Step 2: Use property of tangents
Given: \(FEG\) is tangent to the circle at \(E\), and line \(CE\) meets the circle at \(E\) \[ \angle GEC = 52^\circ \] From circle geometry: \[ \angle GEC = \angle EAC = 52^\circ \quad \text{(angle between tangent and chord equals angle in alternate segment)} \] Step 3: In triangle \(AEC\) \[ \angle AEC = 90^\circ, \quad \angle EAC = 52^\circ \Rightarrow \angle ACE = 180^\circ - 90^\circ - 52^\circ = 38^\circ \] Now: \[ \angle E = \angle EAC = 52^\circ,\quad \angle C = \angle ACE = 38^\circ \Rightarrow \angle E + \angle C = 52^\circ + 104^\circ = \boxed{156^\circ} \]