Quadrilaterals & Parallelograms: The Family Tree of Four-Sided Shapes
Square, rectangle, rhombus, parallelogram, trapezoid, kite — they're all related, and the relationship is the test. Learn the hierarchy and you'll never miss a 'must be / could be' question.
One family, six members
The Texas CBE doesn't ask you to draw quadrilaterals. It asks: “A quadrilateral has [property X]. What is it?” or “Which property does a rectangle always have?” The whole topic comes down to one diagram — the family tree.
A square inherits every property of every box above it: parallelogram + rectangle + rhombus. That's why “every square is a rectangle” is true, but “every rectangle is a square” is false.
Properties cheat sheet
- Parallelogram
- 2 pairs of parallel sides. Opposite sides equal, opposite angles equal, diagonals bisect each other.
- Rectangle
- A parallelogram with four 90° angles. Diagonals are equal.
- Rhombus
- A parallelogram with four equal sides. Diagonals are perpendicular and bisect angles.
- Square
- Both rectangle & rhombus. All four sides equal AND all four angles 90°. Diagonals equal AND perpendicular.
- Trapezoid
- Exactly one pair of parallel sides (the bases). NOT a parallelogram.
- Kite
- Two pairs of consecutive equal sides. One diagonal bisects the other; diagonals are perpendicular.
The diagonals tell the story
Which shape from the property?
A quadrilateral has opposite sides that are both parallel and congruent. What type of quadrilateral must it be?
Open the question →4 congruent sides — what is it?
A quadrilateral has 4 congruent sides. It must be a:
Open the question →The diagonals of a rectangle
The diagonals of a rectangle are always ___.
Open the question →“Must be” vs “could be”
“Must be” = applies to every case (most restrictive label). “Could be” = at least one case fits (any compatible label is correct). A square could be called a rectangle (true). A rectangle must be a square (false — only sometimes).
3-second recap
- Square = rectangle + rhombus + parallelogram all in one
- Rectangle → equal diagonals
- Rhombus → perpendicular diagonals
- Trapezoid → exactly one pair of parallel sides; not a parallelogram
- Read “must be” vs “could be” before picking an answer