In the given figure, O is the centre of the circle, AE is the diameter and ∠AOC = 100°.

The value of ∠CDE + ∠CEA is:

Correct Option: 90°

Given in the figure:

• O is the centre of the circle.

• AE is the diameter of the circle.

• $\angle \text{AOC}=100°$


Step 1: Find $\angle \text{ABC}$ or relation using the central angle.

The angle subtended by arc AC at the centre is $\angle \text{AOC}=100°$.

The angle subtended by the major arc AC at any point on the circumference (like point D or point B) can be found using circle properties.

Angle subtended by arc AC at point D on the circumference:

$\angle \text{ADC}=\frac{1}{2}\angle \text{AOC}=\frac{1}{2}\times 100°=50°$

Step 2: Find $\angle \text{COE}$.

Since AOE is a straight line diameter, the sum of angles on a straight line is 180°:

$\angle \text{AOC}+\angle \text{COE}=180°$

$100°+\angle \text{COE}=180°$

$\angle \text{COE}=80°$

Step 3: Find $\angle \text{CEA}$.

The angle subtended by arc AC at the circumference point E is:

$\angle \text{AEC}=\frac{1}{2}\angle \text{AOC}=\frac{1}{2}\times 100°=50°$

So, $\angle \text{CEA}=50°$.

Step 4: Find $\angle \text{CDE}$.

The angle subtended by arc EC at the circumference point D is half the central angle $\angle \text{COE}$:

$\angle \text{CDE}=\frac{1}{2}\angle \text{COE}=\frac{1}{2}\times 80°=40°$

Step 5: Calculate $\angle \text{CDE}+\angle \text{CEA}$.

$\angle \text{CDE}+\angle \text{CEA}=40°+50°=90°$

Thus, the value of $\angle \text{CDE}+\angle \text{CEA}$ is 90°.