A circle, with centre O, circumscribs a pentagon ABCDE. IF `AB- BC= CD and angle BCD = 126^(@),` find:
`angle AEB`

Answer

To solve the problem step by step, we need to analyze the given information about the pentagon ABCDE and the angles involved.

### Step 1: Understand the given information
We know that:
- A circle with center O circumscribes the pentagon ABCDE.
- \( AB - BC = CD \)
- \( \angle BCD = 126^\circ \)

### Step 2: Establish relationships in triangles
Since O is the center of the circle, the segments OB, OC, and OD are radii of the circle. Therefore:
- \( OB = OC = OD \)

### Step 3: Analyze triangle BOC and COD
From the information given:
- \( AB - BC = CD \) implies \( BC = AB - CD \).
- We can also see that triangles BOC and COD share side OC.

### Step 4: Use the angle BCD to find angles OCB and OCD
By the property of angles in a circle, we have:
- \( \angle OCB = \angle OCD = \frac{1}{2} \angle BCD \)
- Substituting the value of \( \angle BCD \):
\[ \angle OCB = \angle OCD = \frac{1}{2} \times 126^\circ = 63^\circ \]

### Step 5: Find angle OBC
In triangle BOC, since OB = OC (radii of the circle), we have:
- \( \angle OCB = \angle OBC \)
- Therefore, \( \angle OBC = 63^\circ \).

### Step 6: Use the angle sum property in triangle BOC
The sum of angles in triangle BOC is:
\[ \angle OBC + \angle OCB + \angle BOC = 180^\circ \]
Substituting the known values:
\[ 63^\circ + 63^\circ + \angle BOC = 180^\circ \]
This simplifies to:
\[ \angle BOC = 180^\circ - 126^\circ = 54^\circ \]

### Step 7: Find angle AOB
Since \( AB = BC = CD \), we can conclude:
- \( \angle AOB = \angle BOC = \angle COD = 54^\circ \).

### Step 8: Find angle AEB
Using the property that the angle subtended by an arc at any point on the circumference is half the angle subtended at the center:
\[ \angle AEB = \frac{1}{2} \angle AOB \]
Substituting the value of \( \angle AOB \):
\[ \angle AEB = \frac{1}{2} \times 54^\circ = 27^\circ \]

### Final Answer
Thus, the measure of angle AEB is:
\[ \angle AEB = 27^\circ \]