Given information:
- In quadrilateral ABCD, angle a: angle c = 1:3
- In quadrilateral ABCD, angle b: angle d = 5:6
- Angle a + Angle c = 140

Solution:

Step 1: Understanding the given information

From the given information, we can deduce the following:
- In quadrilateral ABCD, the ratio of angle a to angle c is 1:3. This means that angle a is one-fourth of angle c.
- In quadrilateral ABCD, the ratio of angle b to angle d is 5:6. This means that angle b is five-elevenths of angle d.
- Angle a + Angle c = 140. This implies that the sum of angles a and c is equal to 140 degrees.

Step 2: Finding the values of angle a and angle c

Let's assume the value of angle c as 3x degrees.
Therefore, the value of angle a would be x degrees (since angle a is one-fourth of angle c).

According to the given information, angle a + angle c = 140.
Substituting the values of angle a and angle c, we get:
x + 3x = 140
4x = 140
x = 35

Therefore, angle a = x = 35 degrees and angle c = 3x = 3 * 35 = 105 degrees.

Step 3: Finding the values of angle b and angle d

We know that angle b is five-elevenths of angle d.
Let's assume the value of angle d as 11y degrees.
Therefore, the value of angle b would be 5y degrees (since angle b is five-elevenths of angle d).

According to the given information, angle b: angle d = 5:6.
This means that 5y degrees is equal to five parts of the total 11 parts (5 + 6).

Using the concept of proportion, we can set up the following equation:
5y/11y = 5/11

Cross-multiplying, we get:
5 * 11y = 5y * 11
55y = 55y

This equation is true for any value of y.

Therefore, angle b can have any value and angle d would be 11y degrees.

Step 4: Summarizing the values of the angles in quadrilateral ABCD

From our calculations, we have found the following values:
- Angle a = 35 degrees
- Angle b = 5y degrees (where y can be any value)
- Angle c = 105 degrees
- Angle d = 11y degrees (where y can be any value)

Conclusion:

In quadrilateral ABCD, the angles have the following values:
- Angle a = 35 degrees
- Angle b = 5y degrees (where y can be any value)
- Angle c = 105 degrees
- Angle d = 11y degrees (where y can be any value)