Quadrilateral Hierarchy Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quadrilateral Hierarchy.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Four-sided shapes nest inside each other: every square is a rectangle, every rectangle is a parallelogram, because each level just adds a property.

Common stuck point: The procedure for quadrilateral hierarchy is the easy part; the trap is saying 'a rectangle is a square'. Asking "Does the more specific shape have every property of the general one, plus at least one extra constraint?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the more specific shape have every property of the general one, plus at least one extra constraint?

Worked Examples

Example 1

easy
A quadrilateral has three angles of 85°85°, 95°95°, and 110°110°. Find the fourth angle.

Answer

The fourth angle is 70°70°.

First step

1
Step 1: Recall that the sum of interior angles of any quadrilateral is 360°360°.

Full solution

  1. 2
    Step 2: Let the fourth angle be xx. Then 85+95+110+x=36085 + 95 + 110 + x = 360.
  2. 3
    Step 3: 290+x=360290 + x = 360, so x=70°x = 70°.
The interior angles of any quadrilateral (4-sided polygon) sum to 360°. This can be shown by dividing any quadrilateral into two triangles with a diagonal — each triangle contributes 180°, for a total of 360°. This fact applies to all quadrilaterals: squares, rectangles, parallelograms, trapezoids, and irregular quadrilaterals.

Example 2

medium
Explain the quadrilateral hierarchy: How is a square related to a rectangle, rhombus, and parallelogram?

Example 3

medium
A rectangle has length 1212 cm and width 55 cm. What is the length of each diagonal?

Example 4

medium
A quadrilateral has angle measures in the ratio 1:2:3:41:2:3:4. Find the largest angle.

Example 5

medium
A kite has two pairs of consecutive sides of lengths 55 and 88. Find its perimeter.

Example 6

hard
A rhombus has diagonals of length 1616 and 3030. Find its side length.

Example 7

hard
A square has perimeter 3636. Find the length of one of its diagonals.

Example 8

challenge
The midpoints of the sides of any quadrilateral form a parallelogram (Varignon's theorem). If a quadrilateral has diagonals of length 1010 and 1414, find the perimeter of the Varignon parallelogram.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A quadrilateral has all angles equal. What type of quadrilateral is it? What is each angle's measure?

Example 2

hard
In parallelogram ABCDABCD, ∠A=3x+15°\angle A = 3x + 15° and ∠B=5x−5°\angle B = 5x - 5°. Find all four angles. Use the property that consecutive angles in a parallelogram are supplementary.

Example 3

easy
A quadrilateral has angles 90°90°, 80°80°, and 130°130°. Find the fourth angle.

Example 4

easy
Three angles of a quadrilateral are equal and the fourth is 120°120°. Find each of the three equal angles.

Example 5

medium
In parallelogram PQRSPQRS, ∠P=2x+10°\angle P = 2x + 10° and ∠Q=3x+20°\angle Q = 3x + 20°. Find ∠P\angle P.

Example 6

medium
A rhombus has side length 1010 and one angle of 60°60°. Find the length of the shorter diagonal.

Example 7

medium
True or False: A trapezoid is always a parallelogram (use exclusive definition).

Example 8

medium
In rectangle ABCDABCD, the diagonals meet at EE. If AE=2x−1AE = 2x - 1 and BE=x+4BE = x + 4, find ACAC.

Example 9

medium
An isosceles trapezoid has base angles 70°70° and 70°70°. Find the other two angles.

Example 10

medium
A parallelogram has area 4848 square units and base 88. What is its height?

Example 11

hard
In parallelogram ABCDABCD, ∠A:∠B=2:7\angle A : \angle B = 2 : 7. Find ∠C\angle C.

Example 12

hard
A trapezoid has parallel sides of lengths 77 and 1313 and height 44. Find its area.

Example 13

hard
Quadrilateral ABCDABCD has ∠A=3x\angle A = 3x, ∠B=2x+10°\angle B = 2x + 10°, ∠C=4x−10°\angle C = 4x - 10°, ∠D=x+80°\angle D = x + 80°. Find ∠C\angle C.

Example 14

hard
A kite has diagonals 1010 and 2424. Find its area.

Example 15

hard
Which is the most specific name for a quadrilateral with four equal sides and four right angles?

Example 16

challenge
In parallelogram ABCDABCD with AB=8AB = 8 and AD=6AD = 6, the angle bisectors of ∠A\angle A and ∠B\angle B meet at point PP inside the parallelogram. Find ∠APB\angle APB.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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