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

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We have 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. Mark the points A, B, C, and D. Label the angles as given:
- \( \angle BAC = 30^\circ \)
- \( \angle CBD = 70^\circ \)

### Step 3: Use the Property of Angles Subtended by the Same Chord
Since angles subtended by the same chord at the circumference are equal:
- \( \angle BDC \) (which is subtended by chord \( BC \)) is equal to \( \angle BAC \).
- Therefore, \( \angle BDC = 30^\circ \).

### Step 4: Apply the Same Property for the Other Angle
Similarly, since \( \angle CAD \) (which is subtended by chord \( CD \)) is equal to \( \angle CBD \):
- Therefore, \( \angle CAD = 70^\circ \).

### Step 5: Use the Property of Equal Chords
Since \( AD = BC \), the angles subtended by these equal chords at the circumference are equal:
- Thus, \( \angle DBA = \angle BAC = 30^\circ \).

### Step 6: Calculate \( \angle ABC \)
Now, we can find \( \angle ABC \) using the angles we have:
- \( \angle ABC = \angle ABD + \angle CBD \)
- Substituting the known values:
\[ \angle ABC = 30^\circ + 70^\circ = 100^\circ \]

### Final Answer
Thus, the value of \( \angle ABC \) is \( 100^\circ \).

---