3.4 Properties & Proofs of Quadrilaterals

The Quadrilateral Family

Special quadrilaterals form a hierarchy in which each shape inherits every property of the shapes above it. A parallelogram is the foundation, defined by both pairs of opposite sides being parallel. A rectangle, rhombus, and square are all parallelograms with extra constraints, while a trapezoid and a kite sit outside the parallelogram branch. NY Next Gen standard GEO-G.CO.11 requires proving the parallelogram theorems below.

Parallelogram Properties

In every parallelogram:

• Both pairs of opposite sides are parallel and congruent.
• Both pairs of opposite angles are congruent.
• Consecutive (same-side) angles are supplementary, summing to 180 degrees.
• The diagonals bisect each other, cutting one another into equal halves.

Note what is not guaranteed: in a general parallelogram the diagonals are not congruent and not perpendicular, and the angles are not right angles. Those stronger properties define the special parallelograms.

Rectangle, Rhombus, Square

A rectangle adds right angles, which forces its diagonals to be congruent. A rhombus adds four equal sides, which forces its diagonals to be perpendicular and to bisect the vertex angles. A square is the overlap of both: it carries every parallelogram, rectangle, and rhombus property at once, so its diagonals are congruent, perpendicular, and bisect both each other and the angles.

Trapezoids and Kites

A trapezoid has exactly one pair of parallel sides, called the bases; the non-parallel sides are the legs. Its midsegment joins the midpoints of the legs and equals the average of the two bases, (b1 + b2)/2. An isosceles trapezoid has congruent legs and additionally has congruent base angles, congruent diagonals, and a line of symmetry. A kite has two pairs of consecutive congruent sides (not opposite sides); its diagonals are perpendicular, and the main diagonal bisects the other.

Proving a Quadrilateral Is a Parallelogram

There are five accepted ways to prove a quadrilateral is a parallelogram, and any single one is sufficient:

1. Both pairs of opposite sides parallel (the definition).
2. Both pairs of opposite sides congruent.
3. Both pairs of opposite angles congruent.
4. The diagonals bisect each other.
5. One pair of opposite sides is both parallel and congruent.

Upgrading to a Special Type

Once a figure is established as a parallelogram, a single extra fact promotes it:

• Parallelogram + congruent diagonals becomes a rectangle.
• Parallelogram + perpendicular diagonals (or one pair of consecutive sides congruent) becomes a rhombus.
• Parallelogram + diagonals that are congruent and perpendicular becomes a square.

Proving Type on the Coordinate Plane

Coordinate proofs rely on three tools from the reference sheet. Slope: equal slopes mean sides are parallel, and slopes whose product is -1 mean they are perpendicular (a right angle). Distance formula: equal lengths mean sides or diagonals are congruent. Midpoint formula: if the two diagonals share the same midpoint, they bisect each other.

A reliable plan: to prove a parallelogram, show both pairs of opposite sides have equal slopes, or use slope plus distance for the one-pair-parallel-and-congruent shortcut. To prove a rectangle, add that two adjacent sides have slopes multiplying to -1, or that the diagonals are congruent by the distance formula. To prove a rhombus, show all four sides are congruent by the distance formula. To prove a square, satisfy both the rhombus and rectangle conditions.

Worked example. Quadrilateral ABCD has vertices A(0, 0), B(4, 0), C(5, 3), D(1, 3). Slope of AB is 0 and slope of DC is (3 - 3)/(5 - 1) = 0, so they are parallel. Slope of AD is 3 and slope of BC is (3 - 0)/(5 - 4) = 3, so they are parallel. Both pairs of opposite sides are parallel, so ABCD is a parallelogram. Testing the diagonals, AC is the square root of 34 while BD is the square root of 18, so the diagonals are not congruent and ABCD is not a rectangle.

Classifying From a Proof

When a proof asks for the most specific name, list every property you can establish and take the strongest classification the evidence supports. A quadrilateral with both pairs of opposite sides congruent is a parallelogram; if you additionally show one right angle, upgrade to a rectangle; if you instead show two consecutive sides congruent, upgrade to a rhombus; with both, it is a square.

Always give the most specific correct name, because "parallelogram" earns less credit than "square" when the figure is genuinely a square, yet never claim a stronger type than your evidence supports. For a coordinate rhombus, the cleanest route is four applications of the distance formula showing all sides equal; add one perpendicular-slope check between adjacent sides to promote it to a square.

Common Traps

• Assuming a parallelogram's diagonals are congruent; only rectangles and squares guarantee that.
• Calling a kite a parallelogram; a kite's congruent sides are consecutive, not opposite.
• Using the trapezoid midsegment (average of the bases) when a triangle midsegment (half the side) is intended.