The square

Definition: a square is a quadrilateral that is both a rectangle and a rhombus.

• It is a rectangle, so all its angles are right;
• it is a rhombus, so all its sides are equal.

A square has all the properties of a rectangle and all the properties of a rhombus:

• the sides: all four sides are equal, and opposite sides are parallel;
• the angles: all four angles are right;
• the diagonals:
  • are equal (as in a rectangle);
  • bisect each other (as in every parallelogram);
  • are perpendicular and bisect the square’s angles (as in a rhombus).

Given: $\displaystyle ABCD$ is a square.
Prove: $\displaystyle {\angle BAC}={45^{\circ}}$.

The proof below uses a property of the rhombus (the diagonal bisects the angle) and a property of the rectangle (the angle is right).

In the square $\displaystyle ABCD$, $\displaystyle {AC}={10}$.
The diagonals meet at the point $\displaystyle O$.

• The diagonals are equal and bisect each other, so $\displaystyle {AO}={BO}={5}$;
• the diagonals are perpendicular, so in triangle $\displaystyle AOB$ the angle $\displaystyle O$ is right;
• by Pythagoras’ theorem $\displaystyle {AB^2}={5^2+5^2}={50}$, so $\displaystyle {AB}={\sqrt{50}\approx7.07}$.

A square is both a rectangle and a rhombus. So to prove that a quadrilateral is a square, show that it is a rectangle with a property of a rhombus, or a rhombus with a property of a rectangle.

• Theorem: a rectangle with two equal adjacent sides is a square.
  • A rectangle is a parallelogram, and a parallelogram with two equal adjacent sides is a rhombus.
• Theorem: a rectangle whose diagonals are perpendicular is a square.
  • A parallelogram whose diagonals are perpendicular is a rhombus.
• Theorem: a rhombus with a right angle is a square.
  • A rhombus is a parallelogram, and a parallelogram with a right angle is a rectangle.
• Theorem: a rhombus whose diagonals are equal is a square.
  • A parallelogram whose diagonals are equal is a rectangle.

Given: $\displaystyle ABCD$ is a rhombus, and $\displaystyle {AC}={BD}$.
Prove: $\displaystyle ABCD$ is a square.

The proof below shows that $\displaystyle ABCD$ is a rectangle too.

Each special quadrilateral is a case of a more general one, and has all its properties:

• parallelogram: a quadrilateral with two pairs of parallel opposite sides.
  • rectangle: a parallelogram with a right angle.
  • rhombus: a parallelogram with two equal adjacent sides.
    • square: a rhombus that is also a rectangle.
• Every square is a rectangle, and every square is a rhombus.
• Every rhombus is also a kite: it has two pairs of equal adjacent sides.

$\displaystyle ABCD$ is a square, and its diagonals meet at the point $\displaystyle O$.
Work it out and give the reasons.

$\displaystyle {AC}={12}$.
Find $\displaystyle BO$ and the angle $\displaystyle OBA$.

The perimeter of the square is $\displaystyle 28$.
Find the length of its side, the angle $\displaystyle AOB$ and the angle $\displaystyle OAD$.

Is the quadrilateral surely a square? Write yes or no, and give the reason.

A rhombus whose diagonals are equal.

A quadrilateral whose four sides are all equal and that has a right angle.

A quadrilateral whose diagonals are perpendicular and equal.

A rhombus with two equal adjacent angles.

$\displaystyle ABCD$ is a square.
The point $\displaystyle E$ is on the side $\displaystyle BC$, and the point $\displaystyle F$ on the side $\displaystyle CD$, so that $\displaystyle {BE}={CF}$.
Prove: $\displaystyle {AE}={BF}$.

$\displaystyle ABCD$ is a rectangle, and the diagonal $\displaystyle AC$ bisects the angle $\displaystyle A$: $\displaystyle {\angle BAC}={\angle DAC}$.
Prove: $\displaystyle ABCD$ is a square.