Given, the lines are:

y = x ...........1

y = √2x .......2

From equation 1,

m₁ = 1

and from equation 2,

m₂ = √2

Let θ is the angle between the given lines.

Now tan θ = |m₁ - m₂ |/(1 + m₁ * m₂ )

=> tan θ = |(1 - √2)/(1 + 1 * √2)|

=> tan θ = |(1 - √2)/(1 + √2)|

=> tan θ = |(√2 - 1)/(√2 + 1)|

=> tan θ = |{(√2 - 1)/(√2 + 1)}*{(√2 - 1)/(√2 - 1)}|

=> tan θ = |(√2 - 1)² /(2 - 1)|

=> tan θ = |2 + 1 - 2√2|

=> tan θ = |3 - 2√2|

=> tan θ = ±(3 - 2√2)

=> θ = tan^-1 {±(3 - 2√2)}

 So, the angle between the given lines is tan^-1 {±(3 - 2√2)}