Properties of rhombuses

key notes :

Definition:

  • A rhombus is a quadrilateral with all four sides of equal length.

Basic Properties:

  • All sides are congruent (equal in length).
  • Opposite angles are equal.
  • Diagonals bisect each other at right angles (90°).
  • Diagonals bisect the interior angles.

Special Properties:

  • A rhombus is a parallelogram, meaning its opposite sides are parallel.
  • It has two lines of symmetry (along the diagonals).
  • The sum of any two adjacent angles is 180° (consecutive angles are supplementary).

Diagonals of a Rhombus:

  • They are perpendicular bisectors of each other.
  • They divide the rhombus into four right-angled triangles.
  • If the diagonals are d₁ and d₂, then the area of the rhombus is:

Area = 1/2 × d1 × d2

Difference from a Square:

  • A square is a special rhombus where all angles are 90°.
  • In a general rhombus, angles are not necessarily 90°.

Learn with an example

Quadrilateral QRST is a rhombus. What is ∠PRS?

∠PRS = ____°

Since QRST is a rhombus, opposite angles are congruent, QS bisects ∠RQT, and RT bisects ∠QRS.

So, ∠PQR ≅ ∠PQT ≅ ∠PSR ≅ ∠PST and ∠PRQ ≅ ∠PRS ≅ ∠PTQ ≅ ∠PTS. Specifically ∠PQR = ∠PSR = 36°. Also, since QS and RT are perpendicular, ∠RPS = 90° and △PRS is a right triangle.

This means ∠PRS and ∠PSR are complementary. Set the sum of their measures equal to 90° and plug in ∠PSR = 36° to solve for ∠PRS.

∠PRS + ∠PSR = 90°

∠PRS + 36° = 90° Plug in ∠PSR = 36°

∠PRS = 54° Subtract 36° from both sides

So, ∠PRS = 54°.

Quadrilateral GHIJ is a rhombus. What is ∠JIK?

∠JIK = ______°

Since GHIJ is a rhombus, opposite angles are congruent, GI bisects ∠HIJ, and HJ bisects ∠GHI.

So, ∠IHK ≅ ∠GHK ≅ ∠GJK ≅ ∠IJK and ∠HGK ≅ ∠HIK ≅ ∠JGK ≅ ∠JIK. Specifically ∠IHK = ∠IJK = 31°. Also, since GI and HJ are perpendicular, ∠IKJ = 90° and △IJK is a right triangle.

This means ∠IJK and ∠JIK are complementary. Set the sum of their measures equal to 90° and plug in ∠IJK = 31° to solve for ∠JIK.

∠IJK + ∠JIK = 90°

31° + ∠JIK = 90° Plug in ∠IJK = 31°

∠JIK = 59° Subtract 31° from both sides

So, ∠JIK = 59°.

Quadrilateral ABCD is a rhombus. What is ∠ACD?

∠ACD= _____°

Since ABCD is a rhombus, ∠BCD and ∠ABC are supplementary.

Set the sum of their measures equal to 180° and plug in ∠ABC to solve for ∠BCD.

∠ABC + ∠BCD = 180°

62° + ∠BCD = 180° Plug in ∠ABC = 62°

∠BCD = 118° Subtract 62° from both sides

Also, AC bisects ∠BCD, so ∠ACD = ∠ACB = 1/2 ∠BCD. Next, plug in ∠BCD = 118° to this equation and solve for ∠ACD.

∠ACD = 1/2 . ∠BCD

= 1/2 (118°) Plug in ∠BCD = 118°

= 59° Multiply

So, ∠ACD=59°.

Let’s practice!