ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If `∠DBC=70^@` , `∠BAC` is `30^@` , find ∠ BCD. Further if AB=BC, find ∠ ECD.

Options

A. A. 80* and 50*
B. B. 30* and 80*
C. C. 100* and 70*
D. D. none of these

Answer

To solve the problem step by step, we will follow the given information about the cyclic quadrilateral ABCD, where the diagonals intersect at point E. We are given the angles ∠DBC = 70° and ∠BAC = 30°. We need to find ∠BCD and then ∠ECD if AB = BC.

### Step 1: Understand the properties of cyclic quadrilaterals
In a cyclic quadrilateral, the opposite angles are supplementary. This means that the sum of the measures of opposite angles equals 180°.

### Step 2: Identify the angles
We are given:
- ∠DBC = 70° (which is the same as ∠BDC since they are the same angle)
- ∠BAC = 30°

### Step 3: Find ∠BAD
Using the property of cyclic quadrilaterals, we can find ∠BAD. Since ∠BAC and ∠BDC are angles in the same segment, we have:
- ∠BAD = ∠BDC = 70°

### Step 4: Find ∠BCA
Now, we can find ∠BCA using the triangle ABC. Since AB = BC, triangle ABC is isosceles. Therefore, we have:
- ∠BCA = ∠BAC = 30°

### Step 5: Find ∠BDA
Now, we can find ∠BDA using the triangle ABD:
- ∠BDA = ∠BAC + ∠BDC = 30° + 70° = 100°

### Step 6: Find ∠BCD
Using the property of cyclic quadrilaterals again, we know:
- ∠BAD + ∠BCD = 180°
Thus, we can calculate ∠BCD:
- ∠BCD = 180° - ∠BDA = 180° - 100° = 80°

### Step 7: Find ∠ECD
Since AB = BC, we have already established that triangle ABC is isosceles. Therefore, we can find ∠ECD as follows:
- ∠ECD = ∠BCD - ∠BCA = 80° - 30° = 50°

### Final Answers
- ∠BCD = 80°
- ∠ECD = 50°