Properties of Quadrilaterals

A quadrilateral is any polygon with exactly four sides, four vertices, and four interior angles. Quadrilaterals are everywhere — windows, doors, floor tiles, tabletops, picture frames, and roof trusses. Understanding their properties helps you identify shapes, solve for missing measurements, and verify that structures are built correctly.

Interior Angle Sum

Every quadrilateral has interior angles that add up to $360\degree$.

$\text{Sum of interior angles} = 360\degree$

Why? Draw a diagonal across any quadrilateral and you split it into two triangles. Each triangle has angles summing to $180\degree$, so the total is $2 \times 180\degree = 360\degree$.

This rule applies to every quadrilateral — no matter how irregular the shape. If you know three of the four angles, you can always find the fourth.

The Quadrilateral Family

Quadrilaterals form a classification hierarchy where each special type inherits all the properties of the types above it:

• Quadrilateral (most general — 4 sides, angles sum to $360\degree$)
  • Trapezoid (at least one pair of parallel sides)
    • Parallelogram (two pairs of parallel sides)
      • Rectangle (parallelogram with four right angles)
      • Rhombus (parallelogram with four equal sides)
        • Square (rectangle AND rhombus — four right angles AND four equal sides)
  • Kite (two pairs of adjacent equal sides)

Every square is a rectangle, every rectangle is a parallelogram, and every parallelogram is a trapezoid. But the reverse is not true — a trapezoid is not necessarily a parallelogram.

Parallelogram Properties

A parallelogram has two pairs of parallel sides. This single condition produces several useful properties:

• Opposite sides are parallel and equal in length: $AB \parallel CD$ and $AB = CD$; $BC \parallel AD$ and $BC = AD$
• Opposite angles are equal: $\angle A = \angle C$ and $\angle B = \angle D$
• Consecutive angles are supplementary: $\angle A + \angle B = 180\degree$
• Diagonals bisect each other: each diagonal cuts the other into two equal halves

Rectangle Properties

A rectangle is a parallelogram where all four angles are $90\degree$. It has all the properties of a parallelogram, plus:

• All angles are right angles: $\angle A = \angle B = \angle C = \angle D = 90\degree$
• Diagonals are equal in length: $AC = BD$

The diagonals of a rectangle bisect each other (inherited from parallelogram) and are equal in length. This is the property carpenters use to check if a frame is square.

Rhombus Properties

A rhombus is a parallelogram where all four sides are equal. It has all the properties of a parallelogram, plus:

• All sides are equal: $AB = BC = CD = DA$
• Diagonals are perpendicular: the diagonals meet at $90\degree$
• Diagonals bisect the vertex angles: each diagonal splits its vertex angles into two equal parts

A rhombus looks like a “tilted square” — all sides are equal, but the angles are not necessarily $90\degree$.

Square Properties

A square is the most special quadrilateral — it is both a rectangle and a rhombus. It inherits every property from both:

• Four equal sides (from rhombus)
• Four right angles (from rectangle)
• Diagonals are equal (from rectangle)
• Diagonals are perpendicular (from rhombus)
• Diagonals bisect each other (from parallelogram)
• Diagonals bisect the vertex angles (from rhombus)

If someone asks “Is a square a rectangle?” the answer is yes — a square satisfies every condition of a rectangle.

Trapezoid Properties

A trapezoid has at least one pair of parallel sides, called the bases. The non-parallel sides are called the legs. (Under this inclusive definition, parallelograms are special trapezoids — but most everyday trapezoids have only one pair of parallel sides.)

• One pair of parallel sides: the top and bottom (or the two bases)
• Consecutive angles along the same leg are supplementary: they add to $180\degree$

An isosceles trapezoid is a special trapezoid with a line of symmetry perpendicular to the bases (equivalently, its base angles are equal). It has additional properties:

• Base angles are equal: the two angles at each base are the same
• Diagonals are equal in length

Kite Properties

A kite has two pairs of consecutive (adjacent) sides that are equal. It looks like the shape of a flying kite.

• Two pairs of consecutive equal sides: $AB = AD$ and $CB = CD$
• One pair of opposite angles are equal: the angles between the unequal sides are equal
• Diagonals are perpendicular: they meet at $90\degree$
• One diagonal bisects the other: the diagonal connecting the vertices where equal sides meet (the “main axis”) cuts the other diagonal in half

Types of Quadrilaterals

Properties Summary Table

Worked Examples

Example 1: Find the missing angle in a quadrilateral

A quadrilateral has angles of $85\degree$, $110\degree$, and $72\degree$. Find the fourth angle.

Since the interior angles of any quadrilateral sum to $360\degree$:

$\text{Fourth angle} = 360\degree - 85\degree - 110\degree - 72\degree = 93\degree$

Answer: The fourth angle is $93\degree$.

Example 2: Find all angles of a parallelogram

One angle of a parallelogram measures $65\degree$. Find all four angles.

In a parallelogram, opposite angles are equal and consecutive angles are supplementary ($180\degree$).

$\angle A = 65\degree$ $\angle B = 180\degree - 65\degree = 115\degree \quad \text{(consecutive angles are supplementary)}$ $\angle C = 65\degree \quad \text{(opposite to } \angle A\text{)}$ $\angle D = 115\degree \quad \text{(opposite to } \angle B\text{)}$

Check: $65\degree + 115\degree + 65\degree + 115\degree = 360\degree$

Answer: The four angles are $65\degree$, $115\degree$, $65\degree$, and $115\degree$.

Example 3: Identify the quadrilateral from its properties

A quadrilateral has two pairs of parallel sides, all sides equal, and its diagonals are perpendicular. What type of quadrilateral is it?

• Two pairs of parallel sides makes it a parallelogram
• All sides equal makes it a rhombus
• Perpendicular diagonals are consistent with a rhombus (but do not add anything new)

Since the problem does not state that all angles are $90\degree$, it is not necessarily a square.

Answer: The quadrilateral is a rhombus.

Example 4: Find the diagonal of a rectangle

A rectangle measures 9 cm by 12 cm. Find the length of each diagonal.

The diagonal of a rectangle creates a right triangle with the two sides as legs. Using the Pythagorean theorem:

$d = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \text{ cm}$

Both diagonals of a rectangle are equal, so each diagonal is 15 cm.

Answer: Each diagonal is 15 cm.

Example 5: Checking if a door frame is a true rectangle

A carpenter builds a door frame that measures 36 inches wide and 80 inches tall. To check if the frame is a true rectangle, the carpenter measures both diagonals. What should each diagonal measure?

If the frame is a perfect rectangle, both diagonals will be equal. The expected diagonal length is:

$d = \sqrt{36^2 + 80^2} = \sqrt{1296 + 6400} = \sqrt{7696} \approx 87.73 \text{ in}$

The carpenter measures one diagonal at $87\frac{3}{4}$ inches (87.75 in) and the other at $87\frac{3}{4}$ inches. Both diagonals are equal, confirming the frame has true 90-degree corners.

If the diagonals were different lengths, the frame would be a parallelogram (not a rectangle) and the carpenter would need to adjust the corners before hanging the door.

Answer: Each diagonal should measure approximately 87.73 inches (about $87\frac{3}{4}$”). Equal diagonals confirm a true rectangle.

Real-World Application: Carpentry — Checking if a Frame Is Square

The most common use of quadrilateral properties on a job site is the diagonal test for rectangles. Whenever a carpenter builds a wall frame, cabinet box, or door opening, they verify it is a true rectangle by measuring both diagonals.

The rule is simple: if both diagonals are equal, the parallelogram is a rectangle. If they are not equal, the frame is racked (skewed) and needs adjustment.

Here is the step-by-step process:

Step 1 — Build the frame. Assemble the four sides so opposite sides are equal in length. This guarantees a parallelogram.

Step 2 — Measure corner to corner. Use a tape measure to find the length from one corner to the opposite corner. Record the number.

Step 3 — Measure the other diagonal. Measure from the other pair of opposite corners.

Step 4 — Compare. If both measurements match, the frame is a rectangle and all corners are $90\degree$. If one diagonal is longer, the frame is leaning toward the longer diagonal — push that corner inward until the measurements match.

This works because of the rectangle property: a parallelogram with equal diagonals must have all right angles. You don’t need a framing square for every corner — one quick diagonal check verifies the entire frame.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Every quadrilateral has interior angles summing to $360\degree$
• The quadrilateral hierarchy: quadrilateral, trapezoid, parallelogram, rectangle/rhombus, square — each level inherits all properties above it
• A parallelogram has opposite sides parallel and equal, opposite angles equal, and diagonals that bisect each other
• A rectangle adds equal diagonals and four right angles; a rhombus adds four equal sides and perpendicular diagonals
• A square has every property of both a rectangle and a rhombus
• The diagonal test is the fastest way to verify a frame is rectangular: equal diagonals mean all four corners are $90\degree$

Return to Geometry for more topics in this section.