Proof:

Given that in quadrilateral ABCD, the line segments bisecting angle C and angle D meet at point E. We need to prove that angle A is equal to twice angle CED.

Proof:

Step 1: Draw the figure

First, we draw a quadrilateral ABCD. Label the points as shown below:

```
A _______ B
| |
| |
| |
D|______|C
```

Step 2: Construct the bisectors

We construct the bisectors of angle C and angle D. Let the bisector of angle C intersect with the bisector of angle D at point E.

```
A _______ B
| E |
| |
| |
D|______|C
```

Step 3: Analyze the angles

Let's analyze the angles in the quadrilateral ABCD.

- Angle A: It is an exterior angle of triangle AED, so angle A = angle AED.

- Angle B: It is an exterior angle of triangle BEC, so angle B = angle BEC.

- Angle CED: It is an interior angle of triangle AED and triangle BEC. Since the line segments bisect angle C and angle D, angle CED is equal to half of angle C and half of angle D.

Step 4: Prove angle A = 2 * angle CED

To prove angle A = 2 * angle CED, we need to show that angle A = angle CED + angle CED.

From Step 3, we know that angle A = angle AED and angle CED = 1/2 * angle C.

Therefore, we need to show that angle AED = 1/2 * angle C + 1/2 * angle D.

Step 5: Apply angle bisector theorem

By the angle bisector theorem, we know that angle AED is divided into two equal angles by the bisector of angle C, and it is also divided into two equal angles by the bisector of angle D.

Therefore, angle AED = 1/2 * angle C + 1/2 * angle D.

Step 6: Conclusion

From Step 5, we have angle AED = 1/2 * angle C + 1/2 * angle D.

Since angle A = angle AED, we can conclude that angle A = 1/2 * angle C + 1/2 * angle D.

Now, from Step 3, we know that angle CED = 1/2 * angle C.

Therefore, angle A = 1/2 * angle C + 1/2 * angle D = angle CED + angle CED = 2 * angle CED.

Hence, we have proven that angle A is equal to twice angle CED in quadrilateral ABCD.