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5.5 – Rhombi and Squares

Key Terms

  • Adjacent – Next to each other.
    • Two sides or two angles of a figure are adjacent if they are next to each other (side-by-side).
  • Rhombus – A quadrilateral with four congruent sides.
    • The plural of rhombus is rhombi.
    • All rhombi are parallelograms because their opposite sides are parallel and congruent.
  • Square – A quadrilateral with four right angles and four congruent sides.
    • All squares are parallelograms.
    • All squares are rhombi.
    • All squares are rectangles.



  • Polygons
    • All quadrilaterals are polygons.
    • All parallelograms are quadrilaterals, and therefore polygons.
    • All rectangles are parallelograms, and therefore quadrilaterals and polygons.
    • A square is a regular quadrilateral because all its sides and angles are congruent to one another.


  • Alternate Interior Angles
    • When two parallel lines are cut by a transveral, their alternate interior angles are congruent.



  • Rhombi
    • Properties & Rules

GeoB 5.5 Rhombi


    • A rhombus is a parallelogram with four congruent sides.
      • So, if a parallelogram has two adjacent sides that are congruent, then the parallelogram is a rhombus because the opposite sides of a parallelogram are congruent.

GeoB 5.5 Rhombus Diagram


    • Diagonals
      • In a rhombus, the angles where the diagonals intersect always measure 90° (perpendicular).

GeoB 5.5 Rhombi Diagonals Perpen


      • The diagonals of a rhombus divide the interior angle at each vertex into two congruent angles.

GeoB 5.5 Rhombi Diagonals


    • Proof: To prove a parallelogram Is a rhombus:
      • Show that all the sides are congruent.
      • Show that one pair of adjacent sides is congruent.
      • Show that its diagonals are perpendicular to each other.
      • Show that its diagonals bisect each interior angle.


    • Prove: The diagonals of a rhombus intersect at perpendicular angles (right angles).

Geometric Proof Template


  •  Squares
    • A square is a rhombus with four right angles.
    • Squares have all of the properties of quadrilaterals, parallelograms, rhombi, and rectangles.
    • If a parallelogram has four right angles and four congruent sides, then it is a square.
    • The diagonals of a square are always \sqrt{2} times the length of its side.
      • This value (the square root of 2) is known as Pythagoras’s constant. It was the first number proven to be irrational!



  • Quadrilateral Charts and Diagrams

GeoB 5.5 Square Rhombi Venn


GeoB 5.5 Square Rhombi Chart


GeoB 5.5 Quadrilateral Family


  • Summary of Rules
    • All squares are rhombi, but not all rhombi are squares.
    • All squares are rectangles, but not all rectangles are squares.
    • All rhombi and squares are parallelograms.
    • All rhombi and squares are quadrilaterals.
    • Squares and rectangles are quadrilaterals that have congruent consecutive angles (each angle is 90°).
    • Rhombi and squares are the only two quadrilaterals that have diagonals that intersect at 90° angles (their diagonals are perpendicular).
    • The diagonals of a rhombus are perpendicular and bisect each other.


  • Examples
    • If ABCD is a rhombus, then it might be a square.
    • If ABCD is a rhombus, then it MUST be a parallelogram.
    • If PQRS is a parallelogram, then it might be a rhombus.
    • If PQRS is a parallelogram with two adjacent congruent sides, then is MUST be a rhombus.
    • If ZXYW is a square, then it MUST be a rhombus.
    • If MNOP is a rhombus with NO right angles, then it cannot be a square.


More Examples

  • Ex 1. In a given quadrilateral, each side is parallel to its opposite side and the diagonals are not perpendicular. What could it be?
    • Answer: rectangle and parallelogram
    • Reason: rectangles and parallelograms are not always squares.
      • Sometimes they are long and wide, which means the diagonals will not cross at 90° angles.


  • Ex 2. Given that ABCD is a rhombus, what is the value of x?
    • Since we know it is a rhombus, we know the diagonals are perpendicular (90°).
    • We also know that the three angles of a triangle add up to 180°.
    • Setup: (x) + (3x + 7) + (90) = 180
      • Combine like terms: 4x + 97 = 180
      • Subtract 97 on both sides: 4x = 83
      • Divide by 4 on both sides to solve for x
        • x = 20.75°

GeoB 5.5 Example 2


  • Ex 3. When is a rhombus a square?
    • Answer: When its angles are right angles.

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