Special Quadrilaterals Worksheets - Math Monks - Free Printable

Let's solve this Special Quadrilaterals Worksheet step by step. We need to determine which properties apply to each of the listed quadrilaterals: Parallelogram, Rectangle, Square, Trapezoid, Isosceles Trapezoid, and Kite.

We will go through each property and mark an 'X' in the box if that property applies to the given quadrilateral.

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🔷 Definitions Recap

1. Parallelogram:
- Opposite sides are parallel and congruent.
- Opposite angles are congruent.
- Diagonals bisect each other.

2. Rectangle:
- A parallelogram with four right angles.
- All properties of a parallelogram + diagonals are congruent.

3. Square:
- A rectangle with all sides congruent (a rhombus with right angles).
- All properties of a rectangle and a rhombus.

4. Trapezoid:
- Exactly one pair of opposite sides is parallel.

5. Isosceles Trapezoid:
- A trapezoid where non-parallel sides (legs) are congruent.
- Base angles are congruent.
- Diagonals are congruent.

6. Kite:
- Two pairs of adjacent congruent sides.
- One diagonal is a perpendicular bisector of the other.
- One pair of opposite angles are congruent.

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Now, let’s fill out the chart row by row.

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✔ Completed Chart

| Chart | Parallelogram | Rectangle | Square | Trapezoid | Isosceles Trapezoid | Kite |
|------|---------------|-----------|--------|-----------|---------------------|------|
| Four Sides | ✔ | ✔ | ✔ | ✔ | ✔ | ✔ |
| Two congruent sides | ✔ (opposite) | ✔ (opposite) | ✔ (all) | ✘ | ✔ (legs) | ✔ (adjacent pairs) |
| One pair of opposite congruent sides | ✔ | ✔ | ✔ | ✘ | ✔ | ✘ |
| Four congruent sides | ✘ | ✘ | ✔ | ✘ | ✘ | ✘ |
| One pair of opposite parallel sides | ✘ | ✘ | ✘ | ✔ | ✔ | ✘ |
| Diagonals bisect each other | ✔ | ✔ | ✔ | ✘ | ✘ | ✘ |
| Diagonals are perpendicular | ✘ | ✘ | ✔ | ✘ | ✘ | ✔ |
| One pair of opposite congruent angles | ✔ | ✔ | ✔ | ✘ | ✔ | ✔ |
| Four congruent angles | ✘ | ✔ | ✔ | ✘ | ✘ | ✘ |
| All pairs of consecutive angles are supplementary | ✔ | ✔ | ✔ | ✔ (only on the parallel sides) | ✔ | ✘ |
| Two pairs of opposite parallel sides | ✔ | ✔ | ✔ | ✘ | ✘ | ✘ |

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📝 Explanation for Each Row

#### 1. Four Sides
All quadrilaterals have four sides → ✔ for all.

#### 2. Two congruent sides
- Parallelogram: Opposite sides are congruent → ✔
- Rectangle: Same as above → ✔
- Square: All sides equal → ✔
- Trapezoid: No guarantee of congruent sides → ✘
- Isosceles Trapezoid: Legs are congruent → ✔
- Kite: Two pairs of adjacent congruent sides → ✔

> Note: "Two congruent sides" is vague, but since it's not specified as "one pair", we interpret it as having at least two congruent sides → ✔ for all except general trapezoid.

But since trapezoid can have no congruent sides, we put ✘.

#### 3. One pair of opposite congruent sides
- Parallelogram: Both pairs are congruent → ✔ (has one pair)
- Rectangle: Same → ✔
- Square: Same → ✔
- Trapezoid: Not necessarily → ✘
- Isosceles Trapezoid: The legs (non-parallel) are congruent → ✔
- Kite: Adjacent sides are congruent, not opposite → ✘

→ So only parallelogram, rectangle, square, isosceles trapezoid → ✔

#### 4. Four congruent sides
Only square has all four sides equal → ✔

#### 5. One pair of opposite parallel sides
This is the definition of a trapezoid → ✔ for trapezoid and isosceles trapezoid (since it's a type of trapezoid)

Others have two pairs or none → ✘

#### 6. Diagonals bisect each other
- Parallelogram: ✔
- Rectangle: ✔
- Square: ✔
- Trapezoid: ✘
- Isosceles Trapezoid: ✘
- Kite: ✘ (only one diagonal is bisected)

→ Only parallelograms and their special cases (rectangle, square)

#### 7. Diagonals are perpendicular
- Parallelogram: Not necessarily → ✘
- Rectangle: No → ✘
- Square: Yes → ✔
- Trapezoid: No → ✘
- Isosceles Trapezoid: No → ✘
- Kite: Yes → ✔ (one diagonal is perpendicular to the other)

→ ✔ for square and kite

#### 8. One pair of opposite congruent angles
- Parallelogram: Opposite angles are congruent → ✔
- Rectangle: All angles 90° → ✔
- Square: Same → ✔
- Trapezoid: Not necessarily → ✘
- Isosceles Trapezoid: Base angles are congruent → so one pair of opposite angles? Actually, base angles are adjacent to the same base → they are congruent, but not opposite.

Wait — in isosceles trapezoid:
- Angles adjacent to each base are congruent.
- So ∠A ≅ ∠D and ∠B ≅ ∠C (if AB and CD are bases), so opposite angles may not be congruent unless it's a rectangle.

Actually, in isosceles trapezoid, opposite angles are not necessarily congruent.

But base angles are congruent → e.g., ∠A ≅ ∠B (adjacent), but opposite angles are supplementary, not necessarily congruent.

So no one pair of opposite congruent angles?

Wait — let's clarify:

In kite: One pair of opposite angles are congruent (the ones between unequal sides).

In parallelogram: Opposite angles are congruent → ✔

In rectangle: All angles 90° → ✔

In square: ✔

In isosceles trapezoid: No, opposite angles are not congruent. They are supplementary.

But actually, in some sources, kite has one pair of opposite angles congruent, and parallelograms have both pairs.

So:

- Parallelogram: ✔ (both pairs)
- Rectangle: ✔
- Square: ✔
- Trapezoid: ✘
- Isosceles Trapezoid: ✘ (opposite angles are supplementary, not congruent)
- Kite: ✔ (one pair of opposite angles)

✔ So: Parallelogram, Rectangle, Square, Kite

#### 9. Four congruent angles
Only rectangle and square → both have 90° angles → ✔

#### 10. All pairs of consecutive angles are supplementary
- In any quadrilateral with one pair of parallel sides, the consecutive angles between the parallel sides are supplementary.
- Parallelogram: ✔ (because opposite sides parallel)
- Rectangle: ✔
- Square: ✔
- Trapezoid: ✔ (only if the non-parallel sides are not affecting — but in a trapezoid, each pair of angles along the same leg are supplementary → yes, consecutive angles between the parallel sides are supplementary)
- Isosceles Trapezoid: ✔ (same reason)
- Kite: ✘ (not necessarily)

So: ✔ for parallelogram, rectangle, square, trapezoid, isosceles trapezoid

Note: This is true for any trapezoid because of the parallel sides.

#### 11. Two pairs of opposite parallel sides
Only parallelogram, rectangle, square → ✔

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✔ Final Answer (Filled Table):

| Chart | Parallelogram | Rectangle | Square | Trapezoid | Isosceles Trapezoid | Kite |
|-------|---------------|-----------|--------|-----------|---------------------|------|
| Four Sides | ✔ | ✔ | ✔ | ✔ | ✔ | ✔ |
| Two congruent sides | ✔ | ✔ | ✔ | ✘ | ✔ | ✔ |
| One pair of opposite congruent sides | ✔ | ✔ | ✔ | ✘ | ✔ | ✘ |
| Four congruent sides | ✘ | ✘ | ✔ | ✘ | ✘ | ✘ |
| One pair of opposite parallel sides | ✘ | ✘ | ✘ | ✔ | ✔ | ✘ |
| Diagonals bisect each other | ✔ | ✔ | ✔ | ✘ | ✘ | ✘ |
| Diagonals are perpendicular | ✘ | ✘ | ✔ | ✘ | ✘ | ✔ |
| One pair of opposite congruent angles | ✔ | ✔ | ✔ | ✘ | ✘ | ✔ |
| Four congruent angles | ✘ | ✔ | ✔ | ✘ | ✘ | ✘ |
| All pairs of consecutive angles are supplementary | ✔ | ✔ | ✔ | ✔ | ✔ | ✘ |
| Two pairs of opposite parallel sides | ✔ | ✔ | ✔ | ✘ | ✘ | ✘ |

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✔ Summary of Key Points:

- Parallelogram: Has opposite sides/angles congruent, diagonals bisect, etc.
- Rectangle: All angles 90°, diagonals congruent.
- Square: All sides and angles equal, diagonals perpendicular and bisect.
- Trapezoid: One pair of parallel sides.
- Isosceles Trapezoid: Non-parallel sides congruent, base angles congruent.
- Kite: Two pairs of adjacent sides congruent, one diagonal perpendicular, one pair of opposite angles congruent.

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