An elastic collision occurs between two air hockey pucks in which one puck is at rest and the other is moving with a speed of 0.7 m/s. After the collision, the puck initially in motion makes an angle of 20.00 o with its original direction, and the struck puck moves at an angle of 70.00 o on the other side of the original direction. What is the final speed of the first puck? What is the speed of the second?

You can start out with conservation of monentum in x, y directions. That will give you two equations, two unknowns (both puck speeds after the collision)

You should be able to solve it with those two equations.
And,you have the conservation of energy equation if you need a third equation. There is a bit of algebra in all this.