Step 1: Understanding the Question.
A square PQRS is cut into two pieces along an irregular staircase-like line. Panel I shows one piece, and we need to find which of the four shapes in Panel II is the exact matching second piece, so that putting the two pieces together rebuilds the full square with no gaps and no overlaps.
Step 2: Key Formula or Approach.
For two pieces to form a perfect square, every notch (an inward step) on one piece must be filled by a matching tooth (an outward step) on the other piece, and the overall size of the combined pieces must equal the size of square PQRS. So we mentally place each candidate shape next to Panel I along the cut edge and check whether the steps interlock cleanly.
Step 3: Detailed Explanation.
Panel I has a boundary that steps down from corner S to corner Q in a staircase pattern, leaving specific notches cut out of the square.
Options (i) and (ii) both show a shape closer to the letter H, with teeth running along the top that do not follow the same staircase sequence as the cut in Panel I. Placing either of these next to Panel I leaves gaps at some steps and overlaps at others, so neither completes the square.
Options (iii) and (iv) both show a zigzag staircase shape, which is the correct family of shape to expect as the complementary piece. However, only one of the two has its steps running in the right direction and at the right offset to lock into the notches left by Panel I.
Checking (iv), its steps are offset by one unit compared to what Panel I needs, so a step in (iv) does not sit flush against the matching notch in Panel I.
Checking (iii), each step in this shape sits exactly against the corresponding notch in Panel I, so the two pieces together retrace the straight boundary of square PQRS with no gap or overlap.
Step 4: Final Answer.
Shape (iii) is the piece that exactly completes square PQRS.
\[ \boxed{\text{(iii)}} \]
