To solve the problem, we need to apply the geometric property of a quadrilateral with an incircle (a circle that touches all four sides of the quadrilateral).
1. Understanding the In-circle Property: For a quadrilateral that has an incircle (i.e., a circle that touches all four sides), the sum of lengths of opposite sides is equal. This is a known property of a tangential quadrilateral.
2. Applying the Property: If a circle touches the sides of quadrilateral $ABCD$ at points $P$, $Q$, $R$, and $S$ as shown in the figure, then:
$ AB + CD = AD + BC $
3. Verifying Options: Option (1) matches the property: $AB + CD = AD + BC$
Final Answer: The correct relation is $ \mathbf{AB + CD = AD + BC} $.
