In a regular pentagon ABCDE inscribed in a circle . Find the ratio between angle ADE and angle ADC.

Answer

To solve the problem of finding the ratio between angle ADE and angle ADC in a regular pentagon ABCDE inscribed in a circle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Pentagon**:
- We have a regular pentagon ABCDE inscribed in a circle. In a regular pentagon, all sides and angles are equal.

2. **Drawing the Diagram**:
- Draw a circle and inscribe the pentagon ABCDE. Mark the center of the circle as O.

3. **Calculating Central Angles**:
- The total angle around point O (the center of the circle) is 360 degrees. Since the pentagon has 5 equal angles at the center, each central angle (like angle AOE) can be calculated as:
\[ \text{Angle AOE} = \frac{360^\circ}{5} = 72^\circ \]

4. **Finding Angle ADE**:
- Angle ADE is subtended by arc AE at the circumference. The angle subtended at the circumference is half of the angle subtended at the center. Therefore:
\[ \text{Angle ADE} = \frac{1}{2} \times \text{Angle AOE} = \frac{1}{2} \times 72^\circ = 36^\circ \]

5. **Finding Angle ADC**:
- Angle ADC is the sum of angles ADB and BDC. Since both angles are subtended by the arcs AB and BC respectively, and each subtended angle at the circumference is also 36 degrees:
\[ \text{Angle ADB} = 36^\circ \quad \text{and} \quad \text{Angle BDC} = 36^\circ \]
- Thus, we can calculate angle ADC as:
\[ \text{Angle ADC} = \text{Angle ADB} + \text{Angle BDC} = 36^\circ + 36^\circ = 72^\circ \]

6. **Finding the Ratio**:
- Now, we can find the ratio of angle ADE to angle ADC:
\[ \text{Ratio} = \frac{\text{Angle ADE}}{\text{Angle ADC}} = \frac{36^\circ}{72^\circ} = \frac{1}{2} \]

7. **Final Answer**:
- Therefore, the ratio of angle ADE to angle ADC is:
\[ \text{Angle ADE : Angle ADC} = 1 : 2 \]