Construct quadrilaterals and explain what conditions leave open

Learning goal: Construct quadrilaterals with stated side and angle conditions, classify their properties and decide whether the same conditions allow different figures. Use freehand planning, measured tools and a constrained digital construction.

A quadrilateral is a polygon with four sides. Conditions may specify side lengths, parallel relationships, right angles or other measures. Before drawing, list which conditions are given and which are not. A convincing-looking picture is not proof that a missing property was required. Checking what remains free is part of the construction task.

Suppose all four sides must be four units long. A square satisfies the condition, but it is not the only possibility. A rhombus with side length four and an angle of 60 degrees also satisfies it; its adjacent angle is 120 degrees. The two figures have the same side-length list but different angle measures, so they are not congruent and are not similar to each other.

The diagram preserves equal drawn side lengths in both figures. The slanted rhombus's vertical height is less than four, even though its sloping side is four. Do not measure vertical height and claim a side-length condition has failed. A side is the boundary segment connecting its endpoints; a perpendicular height measures a different distance.

To construct a rhombus, draw segment AB of length four. From A, construct a ray making 60 degrees with AB and mark D four units along it. Draw a line through B parallel to AD and a line through D parallel to AB; their intersection is C. The opposite sides are parallel and equal, and all four sides are four. A ruler and protractor can establish the angle and lengths; a digital parallel-line tool can enforce the parallel relationships.

Changing the angle at A to 90 degrees produces a square with the same four side lengths. Changing it to another permitted angle produces another rhombus, provided the construction remains a simple nondegenerate quadrilateral. This family shows that equal side lengths alone do not fix the angles of a quadrilateral. A triangle with three given sides behaves differently because those side lengths fix its angles when a triangle exists.

If the conditions require one pair of opposite sides six units long, the other pair four units long and four right angles, the result is a six-by-four rectangle up to rigid motion. If the right-angle condition is removed but opposite sides must remain parallel, various parallelograms with those side lengths are possible. A slanted parallelogram and a rectangle can share side lengths but differ in angle and height.

Four right angles without side lengths also leave many rectangles possible. A two-by-five rectangle and a three-by-seven rectangle both satisfy that condition. Adding that all four sides are equal narrows the family to squares but still leaves size unspecified. Adding a side length of four then fixes a four-by-four square up to translation, rotation or reflection.

Some conditions conflict. A simple plane quadrilateral's interior angles sum to 360 degrees. Four angles of 100 degrees would sum to 400 and cannot form such a quadrilateral. A rectangle cannot have adjacent angles of 60 and 120 because its definition requires right angles. Be explicit about the type and assumptions rather than accepting any four connected segments as a valid solution.

Classification can be inclusive. A square has four right angles, so it is a rectangle; it also has four equal sides, so it is a rhombus. Both rectangles and rhombi have opposite sides parallel and are parallelograms. Describing a figure as a square does not erase those broader properties. Use the stated definitions to explain overlapping categories.

Compare constructions by invariants and differences. Which side lengths stay fixed? Which angles change? Does a diagonal or perpendicular height change even though the sides do not? A constrained digital construction can make a family visible as a point moves. Record the conditions the tool actually enforces so that an accidental drag does not invalidate the example while you continue claiming it satisfies the original rules.

Worked same sides, different figures: Draw a four-unit square and a four-unit rhombus with one 60-degree angle. Both have four equal sides, but the square has four 90-degree angles and the other alternates 60 and 120. The shared side condition therefore allows different quadrilaterals.

Worked added condition: Start with all sides four. Add four right angles, and the construction must be a square. The added angle condition removes the slanted alternatives; it does not change the side lengths. Explain how a new constraint narrows possibilities rather than merely drawing a new example.

Worked impossible list: Angles 70, 80, 90 and 100 sum to 340, not 360. They cannot be the interior angles of a simple plane quadrilateral. Check the sum before attempting to force a construction to close.

Required construction sequence: Make freehand sketches for the four-equal-side family. Construct the square and 60-degree rhombus with ruler and protractor, checking side lengths and opposite-side parallelism. Reproduce a rhombus in a digital geometry tool using constraints, then vary an angle while retaining equal sides. Record at least two valid examples and one property that changes. If the digital construction is unavailable, identify that component as still pending.

Independent performance: Create four sets of quadrilateral conditions: one that fixes a rectangle up to rigid motion, one that permits several rhombi, one that leaves a family of rectangle sizes and one impossible set. Supply measured drawings or a geometric contradiction for each. Explain the resulting types using their properties and distinguish intended exact conditions from small drawing errors.