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

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We are given a cyclic quadrilateral ABCD where:
- \( AD = BC \)
- \( \angle BAC = 30^\circ \)
- \( \angle CBD = 70^\circ \)

### Step 2: Draw the Diagram
Draw a cyclic quadrilateral ABCD with points A, B, C, and D on the circumference of a circle. Mark the angles \( \angle BAC \) and \( \angle CBD \) as given.

### Step 3: Use the Angle Made by the Same Segment
Since \( CD \) is a chord of the circle, it creates angles \( \angle CBD \) and \( \angle CAD \) at the circumference. According to the property of cyclic quadrilaterals:
\[ \angle CAD = \angle CBD \]
Thus, we have:
\[ \angle CAD = 70^\circ \]

### Step 4: Calculate \( \angle BAD \)
Now, we can find \( \angle BAD \) using:
\[ \angle BAD = \angle CAB + \angle CAD \]
Substituting the known values:
\[ \angle BAD = 30^\circ + 70^\circ = 100^\circ \]

### Step 5: Use the Property of Cyclic Quadrilaterals
In a cyclic quadrilateral, the sum of opposite angles is \( 180^\circ \). Therefore:
\[ \angle BCD + \angle BAD = 180^\circ \]
Substituting \( \angle BAD \):
\[ \angle BCD + 100^\circ = 180^\circ \]
Solving for \( \angle BCD \):
\[ \angle BCD = 180^\circ - 100^\circ = 80^\circ \]

### Step 6: Use the Equal Chords Property
Since \( AD = BC \), the angles subtended by equal chords at the circumference are equal:
\[ \angle BAC = \angle DCA \]
Thus:
\[ \angle DCA = 30^\circ \]

### Step 7: Find \( \angle BCA \)
Now, we can find \( \angle BCA \) using:
\[ \angle BCA + \angle DCA = \angle BCD \]
Substituting the known values:
\[ \angle BCA + 30^\circ = 80^\circ \]
Solving for \( \angle BCA \):
\[ \angle BCA = 80^\circ - 30^\circ = 50^\circ \]

### Final Answer
Thus, the value of \( \angle BCA \) is:
\[ \angle BCA = 50^\circ \]

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