Quadrilaterals - Tenth Grade Geometry
Introduction to Quadrilaterals
Quadrilateral: A polygon with four sides, four vertices, and four angles
Sum of Interior Angles: Always 360°
Number of Diagonals: 2
Types: Parallelogram, Rectangle, Square, Rhombus, Trapezoid, Kite
Key Property: The sum of all interior angles is 360°
Sum of Interior Angles: Always 360°
Number of Diagonals: 2
Types: Parallelogram, Rectangle, Square, Rhombus, Trapezoid, Kite
Key Property: The sum of all interior angles is 360°
Angle Sum Property of Quadrilaterals:
$$\angle A + \angle B + \angle C + \angle D = 360°$$
For any quadrilateral ABCD:
The sum of all four interior angles equals 360 degrees
General Formula:
$$S = (n - 2) \times 180°$$
For quadrilateral: $n = 4$
$$S = (4 - 2) \times 180° = 2 \times 180° = 360°$$
$$\angle A + \angle B + \angle C + \angle D = 360°$$
For any quadrilateral ABCD:
The sum of all four interior angles equals 360 degrees
General Formula:
$$S = (n - 2) \times 180°$$
For quadrilateral: $n = 4$
$$S = (4 - 2) \times 180° = 2 \times 180° = 360°$$
1. Classify Quadrilaterals
Classification by Parallel Sides:
1. Trapezoid (Trapezium): Exactly ONE pair of parallel sides
2. Parallelogram: TWO pairs of parallel sides (opposite sides parallel)
3. Kite: NO parallel sides (but two pairs of adjacent sides equal)
4. General Quadrilateral: No special properties
1. Trapezoid (Trapezium): Exactly ONE pair of parallel sides
2. Parallelogram: TWO pairs of parallel sides (opposite sides parallel)
3. Kite: NO parallel sides (but two pairs of adjacent sides equal)
4. General Quadrilateral: No special properties
Hierarchy of Quadrilaterals:
General Quadrilateral
↓
Trapezoid (1 pair parallel sides)
↓
Parallelogram (2 pairs parallel sides)
↙ ↘
Rectangle (parallelogram + all right angles) Rhombus (parallelogram + all sides equal)
↘ ↙
Square (rectangle + rhombus = all sides equal + all right angles)
General Quadrilateral
↓
Trapezoid (1 pair parallel sides)
↓
Parallelogram (2 pairs parallel sides)
↙ ↘
Rectangle (parallelogram + all right angles) Rhombus (parallelogram + all sides equal)
↘ ↙
Square (rectangle + rhombus = all sides equal + all right angles)
2. Identify Trapezoids
Trapezoid (Trapezium): A quadrilateral with exactly ONE pair of parallel sides
Bases: The two parallel sides
Legs: The two non-parallel sides
Height: Perpendicular distance between bases
Bases: The two parallel sides
Legs: The two non-parallel sides
Height: Perpendicular distance between bases
Types of Trapezoids
1. Isosceles Trapezoid:
• Legs are equal in length
• Base angles are equal
• Diagonals are equal
• Non-parallel sides are congruent
2. Right Trapezoid:
• Has two right angles (90°)
• One leg is perpendicular to both bases
3. Scalene Trapezoid:
• No special properties
• Legs are not equal
• Legs are equal in length
• Base angles are equal
• Diagonals are equal
• Non-parallel sides are congruent
2. Right Trapezoid:
• Has two right angles (90°)
• One leg is perpendicular to both bases
3. Scalene Trapezoid:
• No special properties
• Legs are not equal
Trapezoid Formulas:
Area:
$$A = \frac{1}{2}(b_1 + b_2) \times h$$
Where:
• $b_1$ and $b_2$ = lengths of the two bases (parallel sides)
• $h$ = height (perpendicular distance between bases)
Perimeter:
$$P = a + b_1 + c + b_2$$
Sum of all four sides
Midsegment (Median):
$$m = \frac{b_1 + b_2}{2}$$
The midsegment is parallel to the bases and equals half the sum of the bases
Area:
$$A = \frac{1}{2}(b_1 + b_2) \times h$$
Where:
• $b_1$ and $b_2$ = lengths of the two bases (parallel sides)
• $h$ = height (perpendicular distance between bases)
Perimeter:
$$P = a + b_1 + c + b_2$$
Sum of all four sides
Midsegment (Median):
$$m = \frac{b_1 + b_2}{2}$$
The midsegment is parallel to the bases and equals half the sum of the bases
3. Find Missing Angles in Quadrilaterals
Example 1: Find missing angle
In quadrilateral ABCD, ∠A = 80°, ∠B = 110°, ∠C = 90°. Find ∠D.
Using angle sum property:
$$\angle A + \angle B + \angle C + \angle D = 360°$$
$$80° + 110° + 90° + \angle D = 360°$$
$$280° + \angle D = 360°$$
$$\angle D = 80°$$
Answer: ∠D = 80°
In quadrilateral ABCD, ∠A = 80°, ∠B = 110°, ∠C = 90°. Find ∠D.
Using angle sum property:
$$\angle A + \angle B + \angle C + \angle D = 360°$$
$$80° + 110° + 90° + \angle D = 360°$$
$$280° + \angle D = 360°$$
$$\angle D = 80°$$
Answer: ∠D = 80°
Example 2: Parallelogram angles
In parallelogram PQRS, ∠P = 70°. Find all other angles.
In parallelogram, opposite angles are equal:
∠R = ∠P = 70°
Consecutive angles are supplementary:
∠Q = 180° - 70° = 110°
∠S = 180° - 70° = 110°
Answer: ∠Q = ∠S = 110°, ∠R = 70°
In parallelogram PQRS, ∠P = 70°. Find all other angles.
In parallelogram, opposite angles are equal:
∠R = ∠P = 70°
Consecutive angles are supplementary:
∠Q = 180° - 70° = 110°
∠S = 180° - 70° = 110°
Answer: ∠Q = ∠S = 110°, ∠R = 70°
4. Properties of Parallelograms
Parallelogram: A quadrilateral with both pairs of opposite sides parallel
Symbol: ◻ or ABCD (with parallel marks)
Key Characteristic: Opposite sides are parallel and equal
Symbol: ◻ or ABCD (with parallel marks)
Key Characteristic: Opposite sides are parallel and equal
Properties of Parallelograms:
1. Opposite Sides:
• Are parallel: $AB \parallel CD$ and $BC \parallel AD$
• Are equal: $AB = CD$ and $BC = AD$
2. Opposite Angles:
• Are equal: $\angle A = \angle C$ and $\angle B = \angle D$
3. Consecutive Angles:
• Are supplementary (add to 180°)
• $\angle A + \angle B = 180°$
• $\angle B + \angle C = 180°$
• $\angle C + \angle D = 180°$
• $\angle D + \angle A = 180°$
4. Diagonals:
• Bisect each other (cut each other in half at midpoint)
• $AO = OC$ and $BO = OD$
5. Area and Perimeter:
$$\text{Area} = \text{base} \times \text{height}$$
$$\text{Perimeter} = 2(a + b)$$
Where $a$ and $b$ are lengths of adjacent sides
1. Opposite Sides:
• Are parallel: $AB \parallel CD$ and $BC \parallel AD$
• Are equal: $AB = CD$ and $BC = AD$
2. Opposite Angles:
• Are equal: $\angle A = \angle C$ and $\angle B = \angle D$
3. Consecutive Angles:
• Are supplementary (add to 180°)
• $\angle A + \angle B = 180°$
• $\angle B + \angle C = 180°$
• $\angle C + \angle D = 180°$
• $\angle D + \angle A = 180°$
4. Diagonals:
• Bisect each other (cut each other in half at midpoint)
• $AO = OC$ and $BO = OD$
5. Area and Perimeter:
$$\text{Area} = \text{base} \times \text{height}$$
$$\text{Perimeter} = 2(a + b)$$
Where $a$ and $b$ are lengths of adjacent sides
5. Proving a Quadrilateral is a Parallelogram
Five Ways to Prove a Quadrilateral is a Parallelogram:
Method 1: Both pairs of opposite sides are parallel
Show: $AB \parallel CD$ and $BC \parallel AD$
Method 2: Both pairs of opposite sides are equal
Show: $AB = CD$ and $BC = AD$
Method 3: Both pairs of opposite angles are equal
Show: $\angle A = \angle C$ and $\angle B = \angle D$
Method 4: Diagonals bisect each other
Show: $AO = OC$ and $BO = OD$ (where O is intersection)
Method 5: One pair of opposite sides is both parallel AND equal
Show: $AB \parallel CD$ AND $AB = CD$
Method 1: Both pairs of opposite sides are parallel
Show: $AB \parallel CD$ and $BC \parallel AD$
Method 2: Both pairs of opposite sides are equal
Show: $AB = CD$ and $BC = AD$
Method 3: Both pairs of opposite angles are equal
Show: $\angle A = \angle C$ and $\angle B = \angle D$
Method 4: Diagonals bisect each other
Show: $AO = OC$ and $BO = OD$ (where O is intersection)
Method 5: One pair of opposite sides is both parallel AND equal
Show: $AB \parallel CD$ AND $AB = CD$
Example: Prove using diagonals
Given: In quadrilateral ABCD, diagonals AC and BD bisect each other at O
Prove: ABCD is a parallelogram
Since diagonals bisect each other:
AO = OC (given)
BO = OD (given)
∠AOB = ∠COD (vertical angles)
By SAS, △AOB ≅ △COD
Therefore AB = CD and AB ∥ CD
Conclusion: ABCD is a parallelogram (one pair of opposite sides parallel and equal)
Given: In quadrilateral ABCD, diagonals AC and BD bisect each other at O
Prove: ABCD is a parallelogram
Since diagonals bisect each other:
AO = OC (given)
BO = OD (given)
∠AOB = ∠COD (vertical angles)
By SAS, △AOB ≅ △COD
Therefore AB = CD and AB ∥ CD
Conclusion: ABCD is a parallelogram (one pair of opposite sides parallel and equal)
6. Properties of Rhombuses
Rhombus: A parallelogram with all four sides equal
Also Called: Diamond, equilateral parallelogram
Key Feature: All sides are congruent
Special Type: A rhombus is a special parallelogram
Also Called: Diamond, equilateral parallelogram
Key Feature: All sides are congruent
Special Type: A rhombus is a special parallelogram
Properties of Rhombus:
All Parallelogram Properties PLUS:
1. All Sides Equal:
$$AB = BC = CD = DA$$
2. Diagonals:
• Bisect each other at right angles (90°)
• Are perpendicular: $AC \perp BD$
• Each diagonal bisects the vertex angles
3. Area Formulas:
Using diagonals:
$$A = \frac{1}{2} d_1 \times d_2$$
Where $d_1$ and $d_2$ are the diagonals
Using base and height:
$$A = \text{base} \times \text{height}$$
4. Perimeter:
$$P = 4s$$
Where $s$ = side length
All Parallelogram Properties PLUS:
1. All Sides Equal:
$$AB = BC = CD = DA$$
2. Diagonals:
• Bisect each other at right angles (90°)
• Are perpendicular: $AC \perp BD$
• Each diagonal bisects the vertex angles
3. Area Formulas:
Using diagonals:
$$A = \frac{1}{2} d_1 \times d_2$$
Where $d_1$ and $d_2$ are the diagonals
Using base and height:
$$A = \text{base} \times \text{height}$$
4. Perimeter:
$$P = 4s$$
Where $s$ = side length
Rhombus vs. Square:
• Rhombus: All sides equal, opposite angles equal (angles may not be 90°)
• Square: All sides equal AND all angles are 90°
• Every square is a rhombus, but not every rhombus is a square
• Rhombus: All sides equal, opposite angles equal (angles may not be 90°)
• Square: All sides equal AND all angles are 90°
• Every square is a rhombus, but not every rhombus is a square
7. Properties of Squares and Rectangles
Rectangle
Rectangle: A parallelogram with all angles equal to 90°
Key Feature: All four angles are right angles
Also: Opposite sides are equal and parallel
Key Feature: All four angles are right angles
Also: Opposite sides are equal and parallel
Properties of Rectangle:
All Parallelogram Properties PLUS:
1. All Angles are Right Angles:
$$\angle A = \angle B = \angle C = \angle D = 90°$$
2. Diagonals:
• Are equal in length: $AC = BD$
• Bisect each other
• Are NOT perpendicular (unless it's a square)
3. Area:
$$A = \text{length} \times \text{width}$$
$$A = l \times w$$
4. Perimeter:
$$P = 2(l + w)$$
$$P = 2l + 2w$$
5. Diagonal Length:
$$d = \sqrt{l^2 + w^2}$$
(Using Pythagorean theorem)
All Parallelogram Properties PLUS:
1. All Angles are Right Angles:
$$\angle A = \angle B = \angle C = \angle D = 90°$$
2. Diagonals:
• Are equal in length: $AC = BD$
• Bisect each other
• Are NOT perpendicular (unless it's a square)
3. Area:
$$A = \text{length} \times \text{width}$$
$$A = l \times w$$
4. Perimeter:
$$P = 2(l + w)$$
$$P = 2l + 2w$$
5. Diagonal Length:
$$d = \sqrt{l^2 + w^2}$$
(Using Pythagorean theorem)
Square
Square: A rectangle with all sides equal (OR a rhombus with all right angles)
Special Property: Regular quadrilateral
Key Features: All sides equal AND all angles 90°
Combination: Has ALL properties of both rectangle AND rhombus
Special Property: Regular quadrilateral
Key Features: All sides equal AND all angles 90°
Combination: Has ALL properties of both rectangle AND rhombus
Properties of Square:
All Rectangle Properties + All Rhombus Properties:
1. All Sides Equal:
$$AB = BC = CD = DA = s$$
2. All Angles are 90°:
$$\angle A = \angle B = \angle C = \angle D = 90°$$
3. Diagonals:
• Are equal in length
• Bisect each other
• Are perpendicular (meet at 90°)
• Bisect the vertex angles (each into 45°)
4. Area:
$$A = s^2$$
Or using diagonals:
$$A = \frac{1}{2} d^2$$
Where $d$ = diagonal length
5. Perimeter:
$$P = 4s$$
6. Diagonal Length:
$$d = s\sqrt{2}$$
All Rectangle Properties + All Rhombus Properties:
1. All Sides Equal:
$$AB = BC = CD = DA = s$$
2. All Angles are 90°:
$$\angle A = \angle B = \angle C = \angle D = 90°$$
3. Diagonals:
• Are equal in length
• Bisect each other
• Are perpendicular (meet at 90°)
• Bisect the vertex angles (each into 45°)
4. Area:
$$A = s^2$$
Or using diagonals:
$$A = \frac{1}{2} d^2$$
Where $d$ = diagonal length
5. Perimeter:
$$P = 4s$$
6. Diagonal Length:
$$d = s\sqrt{2}$$
8. Properties of Kites
Kite: A quadrilateral with two pairs of consecutive (adjacent) sides that are equal
Shape: Looks like a traditional flying kite
Key Feature: Two distinct pairs of adjacent equal sides
NOT a Parallelogram: No parallel sides (usually)
Shape: Looks like a traditional flying kite
Key Feature: Two distinct pairs of adjacent equal sides
NOT a Parallelogram: No parallel sides (usually)
Properties of Kites:
1. Adjacent Sides Equal:
• Two pairs of consecutive sides are equal
• $AB = AD$ and $CB = CD$
2. Diagonals:
• Are perpendicular (meet at 90°)
• One diagonal (the "main diagonal") bisects the other
• The main diagonal bisects the vertex angles
3. Angles:
• One pair of opposite angles are equal
• Angles between unequal sides are equal
4. Area:
$$A = \frac{1}{2} d_1 \times d_2$$
Where $d_1$ and $d_2$ are the lengths of the diagonals
5. Perimeter:
$$P = 2a + 2b$$
Where $a$ and $b$ are the lengths of the two different sides
1. Adjacent Sides Equal:
• Two pairs of consecutive sides are equal
• $AB = AD$ and $CB = CD$
2. Diagonals:
• Are perpendicular (meet at 90°)
• One diagonal (the "main diagonal") bisects the other
• The main diagonal bisects the vertex angles
3. Angles:
• One pair of opposite angles are equal
• Angles between unequal sides are equal
4. Area:
$$A = \frac{1}{2} d_1 \times d_2$$
Where $d_1$ and $d_2$ are the lengths of the diagonals
5. Perimeter:
$$P = 2a + 2b$$
Where $a$ and $b$ are the lengths of the two different sides
Example: Find area of kite
A kite has diagonals of length 12 cm and 8 cm. Find its area.
$$A = \frac{1}{2} d_1 \times d_2$$
$$A = \frac{1}{2} \times 12 \times 8$$
$$A = \frac{1}{2} \times 96$$
$$A = 48 \text{ cm}^2$$
Answer: Area = 48 cm²
A kite has diagonals of length 12 cm and 8 cm. Find its area.
$$A = \frac{1}{2} d_1 \times d_2$$
$$A = \frac{1}{2} \times 12 \times 8$$
$$A = \frac{1}{2} \times 96$$
$$A = 48 \text{ cm}^2$$
Answer: Area = 48 cm²
9. Graph Quadrilaterals and Classify on Coordinate Plane
Steps to Classify Quadrilaterals on Coordinate Plane:
Step 1: Plot the Points
Graph all four vertices
Step 2: Calculate Side Lengths
Use distance formula:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Step 3: Calculate Slopes
Use slope formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Step 4: Check Properties
• Parallel sides: Equal slopes
• Perpendicular sides: Slopes are negative reciprocals ($m_1 \times m_2 = -1$)
• Equal sides: Equal distances
Step 5: Classify
Based on properties found
Step 1: Plot the Points
Graph all four vertices
Step 2: Calculate Side Lengths
Use distance formula:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Step 3: Calculate Slopes
Use slope formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Step 4: Check Properties
• Parallel sides: Equal slopes
• Perpendicular sides: Slopes are negative reciprocals ($m_1 \times m_2 = -1$)
• Equal sides: Equal distances
Step 5: Classify
Based on properties found
Example: Classify quadrilateral
Quadrilateral ABCD has vertices A(0, 0), B(4, 0), C(4, 3), D(0, 3). Classify it.
Side lengths:
AB = 4, BC = 3, CD = 4, DA = 3
Opposite sides equal ✓
Slopes:
AB: $m = 0$ (horizontal)
CD: $m = 0$ (horizontal)
AB ∥ CD ✓
BC: undefined (vertical)
DA: undefined (vertical)
BC ∥ DA ✓
Angles:
All angles are 90° (perpendicular sides)
Answer: RECTANGLE (opposite sides parallel and equal, all angles 90°)
Quadrilateral ABCD has vertices A(0, 0), B(4, 0), C(4, 3), D(0, 3). Classify it.
Side lengths:
AB = 4, BC = 3, CD = 4, DA = 3
Opposite sides equal ✓
Slopes:
AB: $m = 0$ (horizontal)
CD: $m = 0$ (horizontal)
AB ∥ CD ✓
BC: undefined (vertical)
DA: undefined (vertical)
BC ∥ DA ✓
Angles:
All angles are 90° (perpendicular sides)
Answer: RECTANGLE (opposite sides parallel and equal, all angles 90°)
Quadrilateral Properties Summary
| Quadrilateral | Sides | Angles | Diagonals | Parallel Sides |
|---|---|---|---|---|
| Parallelogram | Opposite sides equal | Opposite angles equal | Bisect each other | 2 pairs |
| Rectangle | Opposite sides equal | All 90° | Equal, bisect each other | 2 pairs |
| Rhombus | All sides equal | Opposite angles equal | Perpendicular, bisect each other | 2 pairs |
| Square | All sides equal | All 90° | Equal, perpendicular, bisect each other | 2 pairs |
| Trapezoid | No special property | Varies | Not equal | 1 pair |
| Isosceles Trapezoid | Legs equal | Base angles equal | Equal | 1 pair |
| Kite | 2 pairs adjacent sides equal | 1 pair opposite angles equal | Perpendicular, one bisects other | 0 pairs |
Area Formulas Quick Reference
| Quadrilateral | Area Formula | Variables |
|---|---|---|
| Square | $A = s^2$ or $A = \frac{1}{2}d^2$ | s = side, d = diagonal |
| Rectangle | $A = l \times w$ | l = length, w = width |
| Parallelogram | $A = b \times h$ | b = base, h = height |
| Rhombus | $A = \frac{1}{2}d_1 \times d_2$ or $A = b \times h$ | d₁, d₂ = diagonals; b = base, h = height |
| Trapezoid | $A = \frac{1}{2}(b_1 + b_2) \times h$ | b₁, b₂ = bases, h = height |
| Kite | $A = \frac{1}{2}d_1 \times d_2$ | d₁, d₂ = diagonals |
Perimeter Formulas
| Quadrilateral | Perimeter Formula |
|---|---|
| Square | $P = 4s$ |
| Rectangle | $P = 2(l + w)$ or $P = 2l + 2w$ |
| Parallelogram | $P = 2(a + b)$ |
| Rhombus | $P = 4s$ |
| Trapezoid | $P = a + b_1 + c + b_2$ |
| Kite | $P = 2a + 2b$ |
| General | $P = \text{sum of all sides}$ |
Tests for Parallelograms
| Method | What to Prove | Example |
|---|---|---|
| Method 1 | Both pairs of opposite sides parallel | AB ∥ CD and BC ∥ AD |
| Method 2 | Both pairs of opposite sides equal | AB = CD and BC = AD |
| Method 3 | Both pairs of opposite angles equal | ∠A = ∠C and ∠B = ∠D |
| Method 4 | Diagonals bisect each other | AO = OC and BO = OD |
| Method 5 | One pair opposite sides parallel AND equal | AB ∥ CD AND AB = CD |
Coordinate Plane Tools
| Tool | Formula | Use |
|---|---|---|
| Distance Formula | $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ | Find side lengths |
| Slope Formula | $m = \frac{y_2-y_1}{x_2-x_1}$ | Check parallel/perpendicular |
| Midpoint Formula | $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ | Find center of diagonal |
| Parallel Lines | $m_1 = m_2$ | Same slope |
| Perpendicular Lines | $m_1 \times m_2 = -1$ | Negative reciprocals |
Special Properties Comparison
| Property | Parallelogram | Rectangle | Rhombus | Square |
|---|---|---|---|---|
| Opposite sides parallel | ✓ | ✓ | ✓ | ✓ |
| Opposite sides equal | ✓ | ✓ | ✓ | ✓ |
| All sides equal | ✗ | ✗ | ✓ | ✓ |
| Opposite angles equal | ✓ | ✓ | ✓ | ✓ |
| All angles 90° | ✗ | ✓ | ✗ | ✓ |
| Diagonals bisect each other | ✓ | ✓ | ✓ | ✓ |
| Diagonals equal | ✗ | ✓ | ✗ | ✓ |
| Diagonals perpendicular | ✗ | ✗ | ✓ | ✓ |
Success Tips for Quadrilaterals:
✓ All quadrilaterals have angle sum = 360°
✓ Parallelogram: opposite sides parallel AND equal
✓ Rectangle: parallelogram with all right angles
✓ Rhombus: parallelogram with all sides equal
✓ Square: rectangle + rhombus (all sides equal, all angles 90°)
✓ Trapezoid: exactly ONE pair of parallel sides
✓ Kite: two pairs of adjacent sides equal, diagonals perpendicular
✓ Five ways to prove parallelogram (use any one!)
✓ On coordinate plane: use distance formula for sides, slope for parallel/perpendicular
✓ Diagonal properties are key for classification!
✓ All quadrilaterals have angle sum = 360°
✓ Parallelogram: opposite sides parallel AND equal
✓ Rectangle: parallelogram with all right angles
✓ Rhombus: parallelogram with all sides equal
✓ Square: rectangle + rhombus (all sides equal, all angles 90°)
✓ Trapezoid: exactly ONE pair of parallel sides
✓ Kite: two pairs of adjacent sides equal, diagonals perpendicular
✓ Five ways to prove parallelogram (use any one!)
✓ On coordinate plane: use distance formula for sides, slope for parallel/perpendicular
✓ Diagonal properties are key for classification!