ABCD is a cycle quadrilateral. Diagonals BD and AC intersect each other at E. If `angleBEC=128^(@)` and `angleECD=25^(@)` then what is the measure of `angle BAC` ?

Options

A. `52^(@)`
B. `103^(@)`
C. `93^(@)`
D. `98^(@)`

Answer

To solve the problem, we will follow these steps:

1. **Identify Given Angles**: We are given that \( \angle BEC = 128^\circ \) and \( \angle ECD = 25^\circ \).

2. **Find Angle ECD**: Since angles \( BEC \) and \( ECD \) are on a straight line (they form a linear pair), we can find \( \angle CED \):
\[ \angle CED = 180^\circ - \angle BEC = 180^\circ - 128^\circ = 52^\circ \]

3. **Use Triangle Sum Property**: In triangle \( CED \), the sum of the angles is \( 180^\circ \):
\[ \angle CED + \angle ECD + \angle EDC = 180^\circ \]
Substituting the known values:
\[ 52^\circ + 25^\circ + \angle EDC = 180^\circ \]
\[ \angle EDC = 180^\circ - 77^\circ = 103^\circ \]

4. **Relate Angles in Cyclic Quadrilateral**: In a cyclic quadrilateral, angles subtended by the same arc are equal. Therefore, \( \angle EDC = \angle BAC \):
\[ \angle BAC = \angle EDC = 103^\circ \]

5. **Conclusion**: The measure of \( \angle BAC \) is \( 103^\circ \).

### Summary of Steps:
1. Identify given angles.
2. Calculate \( \angle CED \) using linear pair property.
3. Use triangle sum property to find \( \angle EDC \).
4. Relate \( \angle EDC \) to \( \angle BAC \) using properties of cyclic quadrilaterals.
5. State the final answer.