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.
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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 is a rectangle and a rhombus and a parallelogram. A rectangle is a parallelogram. The arrows point from specific to general.
Read the tree downward
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
Diagonals are the fingerprint. Rectangle → equal. Rhombus → perpendicular. Square → both.
Which shape from the property?
A quadrilateral has opposite sides that are both parallel and congruent. What type of quadrilateral must it be?
📌 **Step 1: Recall quadrilateral properties** • **Parallelogram**: Both pairs of opposite sides are parallel AND congruent • **Rectangle**: Parallelogram with 4 right angles • **Rhombus**: Parallelogram with 4 equal sides • **Trapezoid**: Exactly ONE pair of parallel sides 📌 **Step 2: Match the given properties** "Opposite sides are both parallel and congruent" → This is the definition of a **parallelogram**. 💡 **Tip:** Every rectangle, rhombus, and square IS a parallelogram, but not every parallelogram is a rectangle/rhombus/square.
4 congruent sides — what is it?
A quadrilateral has 4 congruent sides. It must be a:
📌 4 congruent sides = rhombus (or square, which is a special rhombus) Not necessarily a square — angles may not be 90°. 💡 Rhombus: 4 equal sides. Square: 4 equal sides + 4 right angles.
The diagonals of a rectangle
The diagonals of a rectangle are always ___.
📌 Rectangle diagonal properties: • Congruent (equal length) ✓ • Bisect each other ✓ • NOT necessarily perpendicular (that's a rhombus)
“Must be” vs “could be”
Read the question word
“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
Check yourself
Quick check #1
Which property is true for ALL parallelograms?
By definition, a parallelogram has two pairs of parallel sides — and those parallel sides are also equal in length. The other properties apply only to special parallelograms (rectangles, rhombi, squares).
Quick check #2
A quadrilateral has 4 right angles AND 4 equal sides. What is it?
4 right angles alone = rectangle. 4 equal sides alone = rhombus. Both at once = square (which is the intersection of rectangle and rhombus on the family tree).