In a circle with centre O, chords AB and CD intersect inside the circumference at E. Find `angle AEC`. If `angle AOC = 40^(@) and angle BOD = 50^(@)`.

Options

A. `90^(@)`
B. `45^(@)`
C. `30^(@)`
D. `60^(@)`

Answer

To find the angle AEC in the given circle with center O, we will follow these steps:

### Step 1: Identify the angles at the center
We are given:
- Angle AOC = 40°
- Angle BOD = 50°

### Step 2: Use the property of angles at the center and circumference
The angle at the center (AOC) is twice the angle at the circumference (ABC) for the same arc. Therefore:
\[ \text{Angle ABC} = \frac{\text{Angle AOC}}{2} = \frac{40°}{2} = 20° \]

### Step 3: Apply the same property for angle BOD
Similarly, for angle BOD:
\[ \text{Angle BCD} = \frac{\text{Angle BOD}}{2} = \frac{50°}{2} = 25° \]

### Step 4: Use the exterior angle property in triangle BCE
In triangle BCE, we can use the exterior angle property which states that the exterior angle (AEC) is equal to the sum of the two opposite interior angles (EBC and BCE):
\[ \text{Angle AEC} = \text{Angle EBC} + \text{Angle BCE} \]
Substituting the values we found:
\[ \text{Angle AEC} = 20° + 25° = 45° \]

### Final Answer
Thus, the value of angle AEC is:
\[ \text{Angle AEC} = 45° \]

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