ABCD is a cyclic quadrilateral , Sides AB and DC produced meet at point E , whereas sides BC and AD produced meet at point F.
If ` angle DCF : angle F : angle E =3 : 5 : 4 ` find the angles of the cyclic quadrilateral ABCD.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Angles
Given the ratio of angles \( \angle DCF : \angle F : \angle E = 3 : 5 : 4 \), we can express these angles in terms of a variable \( x \):
- Let \( \angle DCF = 3x \)
- Let \( \angle F = 5x \)
- Let \( \angle E = 4x \)

### Step 2: Identify Vertical Angles
Since \( \angle DCF \) and \( \angle DAB \) are vertical angles, we have:
- \( \angle DAB = \angle DCF = 3x \)

### Step 3: Find Angle DCB
Since \( \angle DCF = 3x \) and \( \angle DCB \) is on the same straight line as \( \angle DCF \), we can find \( \angle DCB \):
- \( \angle DCB = 180^\circ - \angle DCF = 180^\circ - 3x \)

### Step 4: Apply the Cyclic Quadrilateral Property
In a cyclic quadrilateral, the sum of opposite angles is \( 180^\circ \). Therefore:
- \( \angle DAB + \angle DCB = 180^\circ \)
- Substituting the angles we have:
\[ 3x + (180^\circ - 3x) = 180^\circ \]
This equation is satisfied, confirming our angles are consistent.

### Step 5: Find Angle AFB
In triangle \( AFB \), the sum of the angles is \( 180^\circ \):
- \( \angle A + \angle B + \angle F = 180^\circ \)
- Substituting the angles:
\[ 3x + 7x + 5x = 180^\circ \]
This simplifies to:
\[ 15x = 180^\circ \]

### Step 6: Solve for x
Now, we solve for \( x \):
\[ x = \frac{180^\circ}{15} = 12^\circ \]

### Step 7: Calculate the Angles
Now we can find the angles of the cyclic quadrilateral:
- \( \angle A = 3x = 3 \times 12^\circ = 36^\circ \)
- \( \angle F = 5x = 5 \times 12^\circ = 60^\circ \)
- \( \angle B = 7x = 7 \times 12^\circ = 84^\circ \)
- \( \angle C = 180^\circ - \angle A = 180^\circ - 36^\circ = 144^\circ \)
- \( \angle D = 180^\circ - \angle B = 180^\circ - 84^\circ = 96^\circ \)

### Final Angles of the Cyclic Quadrilateral
Thus, the angles of the cyclic quadrilateral ABCD are:
- \( \angle A = 36^\circ \)
- \( \angle B = 84^\circ \)
- \( \angle C = 144^\circ \)
- \( \angle D = 96^\circ \)