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

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript:

### Step 1: Understand the Given Information
We have a pentagon ABCDE inscribed in a circle with center O. We know:
- \( AB - BC = CD \)
- \( \angle BCD = 126^\circ \)

### Step 2: Establish Relationships in Triangles
In triangles BOC and COD:
- \( OB = OD \) (both are radii of the circle)
- \( OC = OC \) (common side)
- \( BC = CD \) (from the given information)

### Step 3: Apply SSS Criteria for Similarity
Since we have two pairs of equal sides and a common side, triangles BOC and COD are similar by the SSS (Side-Side-Side) criterion.

### Step 4: Find Angles OCB and OCD
From the similarity of triangles, 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:
\[ \angle OCB = \angle OBC \]
Thus,
\[ \angle OBC = 63^\circ \]

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

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

### Step 8: Find Angle AED
Using the property that the angle subtended by an arc at any point on the circle is half the angle subtended by the same arc at the center:
\[ \angle AED = \frac{1}{2} \times (\angle AOB + \angle BOC + \angle COD) \]
Substituting the values:
\[ \angle AED = \frac{1}{2} \times (54^\circ + 54^\circ + 54^\circ) = \frac{1}{2} \times 162^\circ = 81^\circ \]

### Final Answer
Thus, the measure of \( \angle AED \) is:
\[ \angle AED = 81^\circ \]