The quadrilateral hierarchy organizes four-sided polygons by their properties in a classification tree. Every square is a rectangle, every rectangle is a parallelogram, and every parallelogram is a trapezoid — each level adds constraints like equal sides or right angles.
Think of quadrilaterals as a family tree. The most general is any four-sided shape. Add one pair of parallel sides and you get a trapezoid. Add two pairs and you get a parallelogram. Make the angles right and it becomes a rectangle. Make the sides equal and it becomes a rhombus. A square is the 'royal' member—it has every property: parallel sides, equal sides, and right angles.
Showing a random 20 of 50 problems.
Example 1
challenge
The midpoints of the sides of any quadrilateral form a parallelogram (Varignon's theorem). If a quadrilateral has diagonals of length 10 and 14, find the perimeter of the Varignon parallelogram.
Example 2
easy
How many pairs of parallel sides does a trapezoid have (exclusive definition)?
Example 3
hard
True or False: The diagonals of a kite always bisect each other.
Example 4
medium
A parallelogram has area 48 square units and base 8. What is its height?Area
Example 5
hard
In parallelogram ABCD, ∠A=3x+15° and ∠B=5x−5°. Find all four angles. Use the property that consecutive angles in a parallelogram are supplementary.Find all four angles using supplementary consecutive angles.
Example 6
medium
Connecting the midpoints of any quadrilateral's sides always forms what shape?
Example 7
medium
In a parallelogram, one angle is 70°. Find the other three angles.Parallelogram with one angle 70°: find the other three.
Example 8
easy
Is every rhombus a parallelogram? Answer Yes or No.
Example 9
challenge
The midpoint quadrilateral (Varignon parallelogram) of a quadrilateral is a rectangle. What must be true about the original quadrilateral's diagonals?
Example 10
easy
What is a kite (as a quadrilateral)?
Example 11
challenge
A parallelogram has sides 6 and 8 and one diagonal 10. Show it must be a rectangle.
Example 12
easy
A quadrilateral has all angles equal. What type of quadrilateral is it? What is each angle's measure?All angles are equal — what type of shape is this, and what is each angle?
Example 13
easy
A quadrilateral has three angles of 85°, 95°, and 110°. Find the fourth angle.
Example 14
easy
A quadrilateral has angles 90°, 80°, and 130°. Find the fourth angle.
Example 15
hard
A trapezoid has parallel sides of lengths 7 and 13 and height 4. Find its area.Find the area of the trapezoid.
Example 16
medium
Which property do the diagonals of a rhombus have that a general parallelogram's diagonals do not?
Example 17
medium
In rectangle ABCD, the diagonals meet at E. If AE=2x−1 and BE=x+4, find AC.
Example 18
medium
Which statement is always true?
Example 19
hard
A rhombus has diagonals of length 16 and 30. Find its side length.
Example 20
easy
Name the quadrilateral that has exactly one pair of parallel sides (exclusive definition).