ABCD is a cyclic quadrilateral whose diagonals intersect at E. If `angle`BEA = `80^@`, `angle`DBC = `60^@` and `angle`BCD = `40^@`, which of the following statements is true?

Options

A. BD bisects `angle`ADC
B. AB=BC
C. DA=DC
D. AC bisects `angle`BCD

Answer

To solve the problem, we will analyze the given angles in the cyclic quadrilateral ABCD and determine which statement is true based on the properties of cyclic quadrilaterals and the angles formed by the intersecting diagonals.

### Step-by-Step Solution:

1. **Identify the Given Angles**:
- We have the following angles:
- \( \angle BEA = 80^\circ \)
- \( \angle DBC = 60^\circ \)
- \( \angle BCD = 40^\circ \)

2. **Use the Property of Cyclic Quadrilaterals**:
- In a cyclic quadrilateral, opposite angles are supplementary. Therefore, we can find \( \angle AEB \) since \( \angle BEA \) is given:
\[ \angle AEB = \angle BEA = 80^\circ \]

3. **Determine \( \angle EBC \)**:
- Since \( \angle DBC = 60^\circ \) and \( \angle BCD = 40^\circ \), we can find \( \angle EBC \):
\[ \angle EBC = \angle DBC - \angle BCD = 60^\circ - 40^\circ = 20^\circ \]

4. **Calculate \( \angle ECB \)**:
- Since \( \angle BCD = 40^\circ \) and \( \angle EBC = 20^\circ \), we can find \( \angle ECB \):
\[ \angle ECB = \angle BCD - \angle EBC = 40^\circ - 20^\circ = 20^\circ \]

5. **Determine \( \angle AEC \)**:
- Now, we can find \( \angle AEC \) using the triangle \( ABE \):
\[ \angle AEC = 180^\circ - (\angle BEA + \angle EBC) = 180^\circ - (80^\circ + 20^\circ) = 80^\circ \]

6. **Conclusion**:
- Now we have:
- \( \angle AEB = 80^\circ \)
- \( \angle AEC = 80^\circ \)
- Since \( \angle AEB = \angle AEC \), it implies that line AC bisects angle BCD.

### Final Statement:
Thus, the statement that "AC bisects BCD" is true.