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USA - Geometry Quadrilateral theorems

Properties of rectangle, square, rhombus, kite, trapezoid

20 practice questions 1 video lesson Theory + worked examples

Rectangles, Rhombi, Squares, Kites, and Trapezoids

California Geometry • Standard G-CO.11 • Quadrilateral Theorems

Rectangles, Rhombi, Squares, Kites, and Trapezoids is a topic in Quadrilateral Theorems in the California Common Core State Standards. It is aligned to Standard G-CO.11, which requires students to prove theorems about the properties of special parallelograms and other quadrilaterals.

The special quadrilaterals each add properties: a rectangle has equal diagonals, a rhombus perpendicular diagonals, a square both, and a kite and trapezoid their own.

California Geometry › Quadrilateral Theorems › Rectangles, Rhombi, Squares, Kites, and Trapezoids  —  Standard G-CO.11

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Theory

Special quadrilaterals add properties to the parallelogram:

  • Rectangle — a parallelogram with four right angles; its diagonals are congruent.
  • Rhombus — a parallelogram with four equal sides; its diagonals are perpendicular and bisect the angles.
  • Square — both a rectangle and a rhombus (all four properties).
  • Kite — two pairs of adjacent equal sides; its diagonals are perpendicular.
  • Trapezoid — exactly one pair of parallel sides; an isosceles trapezoid has congruent legs and base angles.
A square is the most special: it is a rectangle, a rhombus, and a parallelogram all at once.
Special quadrilaterals Special quadrilaterals Special quadrilaterals rectangle: 4 right angles, diagonals equal rhombus: 4 equal sides, diagonals ⊥ square: both (rectangle + rhombus) kite: 2 pairs adjacent sides, diagonals ⊥
The defining extras of each special quadrilateral.
Rhombus diagonals A rhombus has four equal sides and diagonals that are perpendicular bisectors of each other. rhombus: diagonals meet at right angles
A rhombus: equal sides and perpendicular diagonals.

Diagonal properties:

\[\text{rectangle: diagonals equal};\quad \text{rhombus/kite: diagonals}\perp\]
rectangle diagonals are equal; rhombus and kite diagonals are perpendicular
Equal diagonals \(\Rightarrow\) rectangle; perpendicular diagonals \(\Rightarrow\) rhombus or kite.

How to use the properties

  1. Classify the quadrilateral from its given features.
  2. Recall the diagonal property (equal, perpendicular, or both).
  3. Apply the relevant equal-side, right-angle, or perpendicular-diagonal fact.
Example 1 — Rectangle diagonals
A rectangle has a diagonal of \(13\). Find the other diagonal.
Solution

The diagonals of a rectangle are congruent.

\(\text{other diagonal}\)\(=\)\(13\)
the other diagonal is 13
Example 2 — Rhombus diagonals
The diagonals of a rhombus meet at what angle?
Solution

The diagonals of a rhombus are perpendicular.

\(\text{angle}\)\(=\)\(90^\circ\)
the diagonals meet at 90 degrees
Example 3 — Square properties
Why is a square both a rectangle and a rhombus?
Solution

A square has four right angles (rectangle) and four equal sides (rhombus), so it inherits both sets of properties.

a square is both a rectangle and a rhombus
Example 4 — Trapezoid
How many pairs of parallel sides does a trapezoid have?
Solution

A trapezoid has exactly one pair of parallel sides (the bases). An isosceles trapezoid also has congruent legs and congruent base angles.

a trapezoid has one pair of parallel sides

Common pitfalls

Rectangle diagonals are equal; rhombus diagonals are perpendicular. A square has both.
A trapezoid has exactly one pair of parallel sides. Two pairs make it a parallelogram.
A kite is not a parallelogram. Its equal sides are adjacent, not opposite.

Frequently asked questions

What is special about a rectangle's diagonals?

They are congruent (equal in length), in addition to bisecting each other.

What is special about a rhombus's diagonals?

They are perpendicular and bisect the vertex angles, as well as bisecting each other.

Why is a square both a rectangle and a rhombus?

It has four right angles (rectangle) and four equal sides (rhombus), so it has every property of both.

How many parallel sides does a trapezoid have?

Exactly one pair. An isosceles trapezoid additionally has congruent legs and base angles.

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