ABCD is a cyclic quadrilateral and `A + B = 2 ( C + D)`. If `angle C gt 30^(@)`, then which of the following is true

Options

A. A)`angle D ge 90^@`
B. B)`angle D lt 90^@`
C. C)`angle D le 90^@`
D. D)`angle D gt 90^@`

Answer

To solve the problem, we need to analyze the given information about the cyclic quadrilateral ABCD and the relationship between its angles.

### Step-by-Step Solution:

1. **Understanding the properties of cyclic quadrilaterals**:
- In a cyclic quadrilateral, the sum of opposite angles is equal to 180 degrees. Therefore, we have:
\[ A + B + C + D = 360^\circ \]

2. **Using the given equation**:
- We are given that:
\[ A + B = 2(C + D) \]
- We can express \(A + B\) in terms of \(C\) and \(D\):
\[ A + B = 2(C + D) \implies A + B = 2C + 2D \]

3. **Substituting into the angle sum equation**:
- From the angle sum property, we can substitute \(A + B\):
\[ 2C + 2D + C + D = 360^\circ \]
- This simplifies to:
\[ 3C + 3D = 360^\circ \]
- Dividing through by 3 gives:
\[ C + D = 120^\circ \]

4. **Analyzing the condition on angle C**:
- We are given that \(C > 30^\circ\). Let's denote \(C\) as \(x\) (where \(x > 30^\circ\)):
\[ D = 120^\circ - C = 120^\circ - x \]

5. **Finding the range for angle D**:
- Since \(C > 30^\circ\), we can substitute this into the equation for \(D\):
\[ D = 120^\circ - x < 120^\circ - 30^\circ = 90^\circ \]
- Therefore, we conclude that:
\[ D < 90^\circ \]

6. **Conclusion**:
- Based on the analysis, if \(C > 30^\circ\), then \(D\) must be less than \(90^\circ\). Thus, the correct answer is:
\[ D < 90^\circ \]