Correct Answer: 55°

Solution :

Given that the two circles touch each other externally at P and AB is a direct common tangent to the two circles, A and B are points of contact and $\angle$PAB = 35°.
When a direct common tangent is drawn to both circles and two circles come into external contact at a certain point, the angle subtended by the direct common tangent at that point is 90°.
The angle between a tangent and the radius at the point of contact is 90°.
Such that, $\angle$APB = 90°.
In $\triangle$PAB,
$\angle$ABP = 90° – $\angle$PAB
⇒ $\angle$ABP = 90° – 35°
⇒ $\angle$ABP = 55°
Hence, the correct answer is 55°.