Same Math › Geometry · grades 9–10 › Properties of quadrilaterals
How do kids learn the properties of quadrilaterals?
Quadrilaterals form a family tree. A parallelogram has opposite sides parallel and equal, opposite angles equal, and diagonals that bisect each other. A rectangle is a parallelogram with right angles (so its diagonals are equal). A rhombus is a parallelogram with four equal sides (so its diagonals are perpendicular). A square is both, so it has every property.
- Parallelogram: the starting set of properties.
- Rectangle and rhombus: each adds one.
- Square: everything above it.
Why schools teach it
From sorting shapes to proving them
Fifth graders classify shapes in a hierarchy (5.G.B.4). Geometry students prove the properties of parallelograms (HSG.CO.C.11) and use them to find missing angles and sides with algebra.
The tree answers the classic trick question. Is a square a rectangle? Yes: it has four right angles. Is a rectangle a square? Not always.
At the kitchen table
What to say, and what to skip
Say this
Is it a parallelogram first?
Then all those properties come along.Which properties does it add?
Right angles? Equal sides?Which angles are equal, and which add to 180°?
Opposite are equal; neighbors add to 180°.What do the diagonals do?
Bisect, equal, or perpendicular?
Skip this
- “A square isn't a rectangle.”It is: a rectangle with equal sides.
- Making all four angles equal in a parallelogram.Only opposite angles are equal; neighbors add to 180°.
- Assuming diagonals are equal in every parallelogram.Only in rectangles (and squares).
See it in action
In parallelogram ABCD, ∠A = (3x + 10)° and ∠C = (5x − 30)°
Opposite angles are equal
3x + 10 = 5x − 30Solve
40 = 2x, x = 20Find the angles
∠A = ∠C = 70°, so ∠B = ∠D = 110°
Neighbors add to 180°: 70° + 110° = 180° ✓.
Free practice sheet
Make a quadrilaterals sheet
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Parents also ask
Where do trapezoids and kites fit?
Outside the parallelogram branch: a trapezoid has one pair of parallel sides, and a kite has two pairs of equal neighboring sides.
Do all textbooks agree on trapezoids?
No. Some say exactly one pair of parallel sides, others at least one. Use the class's definition.
How do you prove a shape is a parallelogram?
Show both pairs of opposite sides parallel or equal, or that the diagonals bisect each other.
Is a rhombus a diamond?
It's the math name for one. A diamond tilted to stand on a corner is still a rhombus.