4.5 Quadrilaterals with Equal Sides: the Rhombus

Is the square the only quadrilateral with four equal sides? Draw two equal sides AD and AB that are not perpendicular, then locate C so that CB = CD = AB (swing equal arcs from B and D). The result is a four-equal-sided figure that is not a square.

Definition: Rhombus A rhombus is a quadrilateral in which all four sides are equal in length.

Deduction 9: a rhombus is a parallelogram

In rhombus ABCD, draw the diagonal BD. Because AB = AD, triangle ABD is isosceles, so its base angles match; the same holds in \triangle CBD. The two triangles are congruent (SSS, since all four sides are equal and BD is shared), which forces the diagonal to make equal alternate angles at B and D. Equal alternate angles mean AB \parallel DC, and the other diagonal similarly gives AD \parallel BC. So every rhombus is a parallelogram, inheriting all its properties: opposite angles equal, adjacent angles supplementary, diagonals bisecting each other.

Rhombus ABCD with diagonal BD; equal alternate angles at B and D show the opposite sides are parallel A B C D equal alternate angles ⇒ AB ∥ DC and AD ∥ BC
Figure 4.9 · Deduction 9: the diagonal BD makes equal alternate angles at B and D, so AB ∥ DC and AD ∥ BC.

Worked Example

A rhombus has one angle of 70^\circ. Find all its angles, and the angles a diagonal makes.

Opposite angles are equal and adjacent angles supplementary, so the angles are 70^\circ,\ 110^\circ,\ 70^\circ,\ 110^\circ. A diagonal drawn from a 70^\circ corner splits the figure into isosceles triangles; in such a triangle a + a + 70 = 180, giving base angles a = 55^\circ. So that diagonal cuts each 110^\circ angle into two 55^\circ halves.

Deduction 10: the diagonals cross at 90^\circ

In rhombus ABCD with diagonals meeting at O, compare \triangle ABO and \triangle CBO: AB = CB (equal sides), BO shared, and AO = CO (the other diagonal is bisected). By SSS they are congruent, so \angle AOB = \angle COB; since these add to a straight angle (180^\circ), each equals 90^\circ. A rhombus’s diagonals are perpendicular, and (being a parallelogram’s diagonals) they bisect each other and bisect the corner angles.

Rhombus ABCD with both diagonals: they bisect each other at right angles and bisect the corner angles A B C D O
Figure 4.10 · Deduction 10: a rhombus’s diagonals bisect each other at 90° and bisect the corner angles.

Where does the square sit? A square has four equal sides, so it is a rhombus; it also has four right angles, so it is a rectangle. The square is exactly the overlap, both rhombus and rectangle.

Properties of a Rhombus 1. All four sides are equal. 2. Opposite sides are parallel. 3. Adjacent angles add to 180^\circ; opposite angles are equal. 4. The diagonals bisect each other. 5. The diagonals bisect the corner angles. 6. The diagonals cross at 90^\circ (they are perpendicular).

Figure it Out: Parallelograms & Rhombuses

Practice

  1. Find the remaining angles.
    1. Parallelogram ABCD with \angle A = 55^\circ.
    2. Parallelogram WXYZ with \angle XYZ = 105^\circ.
    3. Rhombus PQRS in which a diagonal makes a 25^\circ angle (\angle PRS = 25^\circ).
    4. Rhombus EFGH in which a diagonal makes a 40^\circ angle (\angle EGF = 40^\circ).
  2. Using diagonal properties, construct a parallelogram whose diagonals are 8\ \text{cm} and 6\ \text{cm} and cross at 130^\circ.
  3. Using diagonal properties, construct a rhombus whose diagonals are 6\ \text{cm} and 8\ \text{cm}.
  1. (i) \angle B = 125^\circ, \angle C = \angle A = 55^\circ, \angle D = \angle B = 125^\circ (opposite angles equal, adjacent supplementary). (ii) \angle XYZ = 105^\circ, so the opposite angle is 105^\circ and the other two are each 180 - 105 = 75^\circ. (iii) The diagonal bisects the corner, so the full corner angle at that vertex is 2 \times 25 = 50^\circ; opposite angle 50^\circ, and the other two are each 180 - 50 = 130^\circ. (iv) The full corner angle is 2 \times 40 = 80^\circ; opposite angle 80^\circ, and the remaining two are each 180 - 80 = 100^\circ.
  2. Draw a segment AB = 8\ \text{cm}, mark its midpoint O, draw a 130^\circ angle at O, and on that line cut OC = OD = 3\ \text{cm} (half of 6\ \text{cm}). Join AC, AD, BC, BD: ACBD is the parallelogram (diagonals bisect each other but are unequal, so no right angles).
  3. Draw a segment AB = 8\ \text{cm}, mark midpoint O, raise a perpendicular at O, and cut OC = OD = 3\ \text{cm} (half of 6\ \text{cm}). Join AC, BC, BD, AD: because the diagonals bisect each other at 90^\circ, ACBD is a rhombus.