Certainly! This problem consists of completing each figure in different ways to satisfy the conditions (a rectangle, a parallelogram, or a general quadrilateral). Let’s analyze and solve each case:

a) Completing the figure to form a rectangle

The given shape consists of two perpendicular sides, $\overline{AB}$ and $\overline{AD}$, forming a right angle at $A$. To complete the rectangle:

• A rectangle has four right angles.
• We need to ensure that:
  1. $\overline{BC} \parallel \overline{AD}$,
  2. $\overline{CD} \parallel \overline{AB}$.

Thus, there is only one way to complete the figure to form a rectangle. We can uniquely locate $C$ such that these conditions are satisfied.

b) Completing the figure to form a parallelogram

The given shape consists of two sides, $\overline{LM}$ and $\overline{MN}$, connected at $M$. To complete the parallelogram:

• A parallelogram has opposite sides parallel and equal in length.
• We need to ensure that:
  1. $\overline{LN} \parallel \overline{MN}$,
  2. $\overline{LM} \parallel \overline{NO}$ (where $O$ is the missing vertex).

For any choice of $O$ along the line parallel to $\overline{LM}$ and $\overline{LN}$, we get a valid parallelogram. Since $O$ can be freely chosen along this line, there are infinitely many ways to complete the figure as a parallelogram.

c) Completing the figure to form any quadrilateral

The given shape consists of two perpendicular sides, $\overline{MP}$ and $\overline{MN}$, forming a right angle at $M$. To complete the quadrilateral:

• A general quadrilateral has no restrictions on angles or side lengths.
• The fourth point $Q$ can be freely chosen anywhere in the plane (not on the existing lines $\overline{MP}$ or $\overline{MN}$).

Thus, there are infinitely many ways to complete the figure to form a general quadrilateral.

Summary of Results

1. Rectangle (a): 1 unique way.
2. Parallelogram (b): Infinitely many ways.
3. General quadrilateral (c): Infinitely many ways.

Would you like a more detailed breakdown or clarification on any step?

Follow-up Questions

1. What defines a rectangle geometrically, and how does it differ from other quadrilaterals?
2. Why does a parallelogram allow infinitely many completions, unlike the rectangle?
3. How can we mathematically express the conditions for $O$ in the parallelogram case?
4. What additional restrictions would limit the completions for the general quadrilateral?
5. Can this approach generalize to figures with more sides, like pentagons or hexagons?

Tip

To construct accurate diagrams of shapes, use the properties of parallelism and perpendicularity as guiding principles.