Given:
Points A(3, 1), B(5, 1), C(a, b), and D(4, 3) are vertices of a parallelogram ABCD.

To find:
The values of a and b.

Solution:
We know that opposite sides of a parallelogram are equal in length and parallel to each other. Using this property, we can find the values of a and b by considering the distance between the given points.

Step 1: Finding the length of AB:
Using the distance formula, the length of AB can be calculated as follows:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((5 - 3)^2 + (1 - 1)^2)
= sqrt(2^2 + 0^2)
= sqrt(4)
= 2

Step 2: Finding the length of CD:
Using the distance formula, the length of CD can be calculated as follows:
CD = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - a)^2 + (3 - b)^2)

Step 3: Finding the length of AD:
Using the distance formula, the length of AD can be calculated as follows:
AD = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - 3)^2 + (3 - 1)^2)
= sqrt(1^2 + 2^2)
= sqrt(5)

Step 4: Finding the length of BC:
Using the distance formula, the length of BC can be calculated as follows:
BC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((5 - a)^2 + (1 - b)^2)

Now, since opposite sides of a parallelogram are equal in length, we can equate AB and CD, as well as AD and BC.

2 = sqrt((4 - a)^2 + (3 - b)^2) (Equation 1)
sqrt(5) = sqrt((5 - a)^2 + (1 - b)^2) (Equation 2)

Squaring both sides of Equation 1 and Equation 2, we get:

4 = (4 - a)^2 + (3 - b)^2 (Equation 3)
5 = (5 - a)^2 + (1 - b)^2 (Equation 4)

Step 5: Solving the equations:
Expanding the squares in Equation 3 and Equation 4, we get:

4 = (16 - 8a + a^2) + (9 - 6b + b^2)
5 = (25 - 10a + a^2) + (1 - 2b + b^2)

Simplifying the equations, we get:

a^2 - 8a + b^2 - 6b + 25 = 0 (Equation 5)
a^2 - 10a + b^2 - 2b + 21 = 0