Practical Geometry - Construction of Quadrilaterals given different conditions (sides, diagonals, angles)

Fundamental Requirement for Unique Construction: To construct a unique quadrilateral, exactly five independent measurements (elements) are required. These can be combinations of sides, diagonals, and angles. For example, knowing only four sides is insufficient because the shape can 'flex' or 'tilt' into different parallelograms unless a fifth measurement, like a diagonal, is provided to lock the structure.

The Role of the Rough Sketch: Before starting the actual construction with a compass and ruler, always draw a freehand rough sketch. Label the vertices in cyclic order (e.g., $A, B, C, D$ going clockwise). This visual map helps you identify which triangles can be constructed first using the SSS (Side-Side-Side) or SAS (Side-Angle-Side) criteria.

Construction using Four Sides and One Diagonal: This is the most common condition where the diagonal divides the quadrilateral into two distinct triangles. Visualize the quadrilateral as two triangles, $\triangle ABC$ and $\triangle ADC$, sharing a common base $AC$. You first construct $\triangle ABC$ using the SSS criterion and then use the remaining two sides to find vertex $D$ relative to the fixed points $A$ and $C$.

Construction using Three Sides and Two Diagonals: When two diagonals and three sides are given, the strategy shifts to constructing a triangle formed by two diagonals and one side. For instance, if you know sides $AB, BC, CD$ and diagonals $AC, BD$, you first construct $\triangle ABC$. Then, using $B$ and $C$ as centers, you draw arcs with radii equal to $BD$ and $CD$ respectively to find the intersection point $D$.

Construction using Two Adjacent Sides and Three Angles: In this case, you must start by drawing the side that has two known angles at its endpoints. For example, if $AB$ is the base and $\angle A$ and $\angle B$ are known, draw $AB$ first, then construct the rays for both angles. Use the second side length to mark a point on one of these rays, and from that new vertex, construct the third given angle to find the final intersection point.

Properties of Special Quadrilaterals: For specific shapes like Parallelograms, Rhombuses, and Squares, you may be given fewer than five measurements because their properties provide the hidden values. For a Parallelogram, opposite sides and angles are equal. For a Rhombus, all sides are equal and diagonals bisect each other at $90^{\circ}$ (right angles). For a Square, all sides are equal and every interior angle is $90^{\circ}$.

Using the Angle Sum Property: If you are given three angles of a quadrilateral, you can always find the fourth angle using the formula $\text{Fourth Angle} = 360^{\circ} - (\text{Sum of three given angles})$. This is particularly useful when the side you want to use as a base doesn't have its corresponding angles explicitly stated in the problem.