In cyclic quadrilateral ABCD , AD = BC `, angle BAC = 30 ^(@) and angle CBD = 70 ^(@) ` , find :
` angle BCD `

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of the cyclic quadrilateral
In a cyclic quadrilateral, the sum of the opposite angles is equal to 180 degrees.

### Step 2: Identify the given information
We are given:
- \( AD = BC \)
- \( \angle BAC = 30^\circ \)
- \( \angle CBD = 70^\circ \)

### Step 3: Draw the diagram
Draw a cyclic quadrilateral ABCD with points A, B, C, and D on the circumference of a circle. Mark the angles as given in the problem.

### Step 4: Find angle CAD
Since angles subtended by the same chord are equal, we have:
\[ \angle CAD = \angle CBD = 70^\circ \]

### Step 5: Calculate angle BAD
Now, we can find angle BAD using the angles in triangle ABC:
\[ \angle BAD = \angle BAC + \angle CAD = 30^\circ + 70^\circ = 100^\circ \]

### Step 6: Use the property of cyclic quadrilaterals
According to the property of cyclic quadrilaterals:
\[ \angle BAD + \angle BCD = 180^\circ \]
Substituting the value of angle BAD:
\[ 100^\circ + \angle BCD = 180^\circ \]

### Step 7: Solve for angle BCD
Now, we can find angle BCD:
\[ \angle BCD = 180^\circ - 100^\circ = 80^\circ \]

### Final Answer
Thus, the measure of angle BCD is:
\[ \angle BCD = 80^\circ \]

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