Rectangle, Rhombus, and Square

Within the family of parallelograms, three special cases dominate: the rectangle (all right angles), the rhombus (all sides equal), and the square (both , all right angles and all sides equal). Each adds one constraint to the parallelogram, and each gets one extra property of its diagonals. This lesson catalogues them carefully.

Definitions

• A rectangle is a parallelogram with at least one right angle. (By the parallelogram angle relations, all four angles are then right angles.)
• A rhombus is a parallelogram with at least one pair of adjacent sides equal. (By the parallelogram property "opposite sides equal", all four sides are then equal.)
• A square is a parallelogram that is both a rectangle and a rhombus , all angles right and all sides equal.

Extra properties of diagonals

Theorem (Rectangle). The diagonals of a rectangle are equal in length.

Proof. In rectangle $ABCD$, consider $\triangle ABC$ and $\triangle BAD$. $AB = BA$ (common), $\angle ABC = \angle BAD = 90^\circ$, and $BC = AD$ (opposite sides of parallelogram). By SAS, $\triangle ABC \cong \triangle BAD$. By CPCT, $AC = BD$. Q.E.D.

Theorem (Rhombus). The diagonals of a rhombus bisect each other at right angles, and they bisect the angles of the rhombus.

Proof sketch. In rhombus $ABCD$, the diagonals already bisect each other (parallelogram property). Consider $\triangle AOB$ and $\triangle AOD$. We have $AB = AD$ (rhombus), $AO = AO$ (common), $OB = OD$ (parallelogram bisection). By SSS, the two triangles are congruent. So $\angle AOB = \angle AOD$, and since they form a linear pair summing to $180^\circ$, each is $90^\circ$. The diagonals are perpendicular. By CPCT, $\angle BAO = \angle DAO$, so the diagonal $AC$ bisects $\angle A$. Similarly for the other diagonal.

Theorem (Square). The diagonals of a square are equal and bisect each other at right angles.

Proof. A square is both a rectangle and a rhombus, so both sets of properties apply.

Properties summary

Recognising a rectangle, rhombus, or square

To prove a quadrilateral is a:

• Rectangle: show it is a parallelogram with one right angle, OR show it has all four angles right, OR show it is a parallelogram with equal diagonals.
• Rhombus: show it is a parallelogram with all four sides equal, OR show it is a parallelogram with perpendicular diagonals, OR show all four sides equal.
• Square: show it is both a rectangle and a rhombus, OR show all sides equal and one angle right.

Worked examples

Example 1. In rectangle $ABCD$, the diagonals meet at $O$. If $AC = 10$, find $OA, OC$.

Diagonals are equal and bisect each other: $OA = OC = 5$.

Example 2. In rhombus $ABCD$, $\angle A = 60^\circ$. Find $\angle B, \angle C, \angle D$.

Opposite angles equal, consecutive supplementary: $\angle C = 60^\circ$, $\angle B = \angle D = 120^\circ$.

Example 3. A square has side $5$. Find its diagonal.

In right triangle $\triangle ABC$ with $AB = BC = 5$ and $\angle B = 90^\circ$, diagonal $AC = \sqrt{5^2 + 5^2} = 5\sqrt{2}$.

Example 4. In a rhombus with diagonals $6$ and $8$, find the side.

Diagonals bisect each other at right angles. Half-diagonals: $3$ and $4$. Side $= \sqrt{3^2 + 4^2} = 5$.

Example 5. Prove: a parallelogram with equal diagonals is a rectangle.

In parallelogram $ABCD$ with $AC = BD$, consider $\triangle ABC$ and $\triangle BAD$. $AC = BD$ (given), $AB = BA$ (common), $BC = AD$ (parallelogram). By SSS, $\triangle ABC \cong \triangle BAD$. By CPCT, $\angle ABC = \angle BAD$. Since these are consecutive angles of the parallelogram, they are supplementary too. So $\angle ABC + \angle BAD = 180^\circ$ and they are equal: each is $90^\circ$. Hence the parallelogram is a rectangle.

Try it yourself

1. State the definition of a rectangle, rhombus, and square.
2. In rectangle $ABCD$, $\angle A = 90^\circ$. State the other three angles.
3. In a rhombus, the diagonals are $\ldots$ (fill in).
4. In a square of side $a$, find the diagonal length.
5. In a rhombus with diagonals $10$ and $24$, find the side.
6. Prove that the diagonals of a rectangle are equal.
7. Is a square a rectangle? A rhombus? Both?
8. Prove that a parallelogram with one right angle is a rectangle.
9. Prove that a parallelogram with perpendicular diagonals is a rhombus.
10. A rhombus has side $5$ and one angle $60^\circ$. Find the diagonals.

Pitfalls / Insight

• A square is both a rectangle and a rhombus. Don't pick just one.
• Rectangles do not have all sides equal in general. Many students assume so by mistake.
• Rhombus diagonals are perpendicular bisectors. Use this when proving rhombus-related claims.

Insight. Each special parallelogram is just "parallelogram + one extra constraint". That constraint always shows up as an additional property of the diagonals. Memorise the table and read off properties as needed.