Three kinds of parallelograms get their own names. A rectangle has four right angles. A rhombus has four congruent sides. A square is both. Each one has every property from Properties of Parallelograms, plus more, and those extra properties give new tests like the ones in Proving a Quadrilateral Is a Parallelogram.
Why it works
Rectangle diagonals are congruent. In rectangle $ABCD$, $\triangle ABC$ and $\triangle DCB$ share $\overline{BC}$, have $AB = DC$, and have right angles at $B$ and $C$. By SAS they are congruent, so $AC = DB$.
Rhombus diagonals are perpendicular. In rhombus $ABCD$ with diagonals meeting at $E$, $\triangle AEB \cong \triangle CEB$ by SSS ($AB = CB$, $AE = CE$, shared $\overline{BE}$). So the angles at $E$ are congruent, and since they form a linear pair, each is $90^\circ$. The same triangles show that each diagonal bisects the angles it passes through.
Rhombus diagonals are perpendicular. In rhombus $ABCD$ with diagonals meeting at $E$, $\triangle AEB \cong \triangle CEB$ by SSS ($AB = CB$, $AE = CE$, shared $\overline{BE}$). So the angles at $E$ are congruent, and since they form a linear pair, each is $90^\circ$. The same triangles show that each diagonal bisects the angles it passes through.
| Shape | Diagonals | Extra test (for a parallelogram) |
|---|---|---|
| rectangle | congruent | congruent diagonals $\Rightarrow$ rectangle |
| rhombus | perpendicular; each bisects a pair of opposite angles | perpendicular diagonals $\Rightarrow$ rhombus |
| square | congruent and perpendicular | both $\Rightarrow$ square |
Key idea
Rectangles have congruent diagonals; rhombuses have perpendicular diagonals that bisect the angles; squares have both. A parallelogram with congruent diagonals is a rectangle, and one with perpendicular diagonals is a rhombus.
Worked example: Rectangle diagonals
$ABCD$ is a rectangle with $AC = 4x - 6$ and $BD = 2x + 10$. Find $AC$.
- Rectangle diagonals are congruent: $4x - 6 = 2x + 10$, so $x = 8$.
- $AC = 4(8) - 6 = 26$.
Answer: $AC = 26$
Worked example: A rhombus side from its diagonals
A rhombus has diagonals of length $10$ and $24$. Find the length of a side.
- The diagonals bisect each other at right angles, making four right triangles with legs $5$ and $12$.
- Each side is a hypotenuse: $\sqrt{5^2 + 12^2} = \sqrt{169} = 13$.
Answer: $13$
Worked example: A rhombus angle
In rhombus $PQRS$, $m\angle QPR = (3x + 2)^\circ$ and $m\angle SPR = (5x - 14)^\circ$. Find $m\angle QPS$.
- Diagonal $\overline{PR}$ bisects $\angle QPS$, so $3x + 2 = 5x - 14$ and $x = 8$.
- Each half is $26^\circ$, so $m\angle QPS = 52^\circ$.
Answer: $52^\circ$
Common mistake
Deciding that any quadrilateral with perpendicular diagonals is a rhombus. A kite has perpendicular diagonals too, and it is not a parallelogram at all. The test "perpendicular diagonals means rhombus" only works once you know the figure is a parallelogram. Check that first.