Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Quadrilaterals ↓


Properties of Parallelograms


A parallelogram is a special type of quadrilateral, a polygon with four sides. Understanding the properties of a parallelogram will help you understand a fundamental concept of geometry that is often used in mathematics for a variety of problems and proofs. A parallelogram is defined by opposite sides that are not only equal in length but also parallel to each other.

Basic definition of a parallelogram

A parallelogram is a four-sided shape (or quadrilateral) where both pairs of opposite sides are parallel. The definition of parallel sides is that they are lines that never meet or intersect, no matter how much you extend them. To represent a parallelogram, we can write it with the parallelogram symbol (//). If the name of the parallelogram is ABCD, you can write it as ABCD or //ABCD.

    //ABCD shows that ABCD is a parallelogram

Properties of parallelogram

1. Opposite sides are equal

One of the most basic properties of a parallelogram is that each pair of opposite sides is equal in length. For a parallelogram //ABCD, this means that:

    BC = DA

2. Opposite angles are equal

Another important property is that opposite angles in a parallelogram are equal. This means that if you look at the angles of a parallelogram //ABCD:

    ∠A = ∠C and ∠B = ∠D

3. Consecutive angles are complementary

Any two consecutive angles in a parallelogram are complementary. This means that the sum of their measures is 180 degrees. For example:

    ∠A + ∠B = 180°
    ∠b + ∠c = 180°
    ∠C + ∠D = 180°
    ∠D + ∠A = 180°

4. Diagonals bisect each other

The diagonals of a parallelogram bisect each other, which means they cut each other exactly in half. So, if AC and BD are the diagonals of a parallelogram //ABCD, then:

    AO = OC and BO = OD

Visual example

Consider the visual illustration below:

A B C D

Here, parallelogram ABCD has sides AB parallel to CD, and BC parallel to DA. Diagonals AC and BD bisect each other.

Formulas related to parallelogram

Area

The area of a parallelogram can be calculated using the following formula:

    Area = base × height

In the context of a parallelogram //ABCD, if AB is considered as the base and the perpendicular altitude from C to AB is considered as the altitude, then this formula helps in determining how much space the parallelogram covers.

Circumference

The perimeter of a parallelogram can be found using the following formula:

    Perimeter = 2 × (base + side)

Using parallelogram //ABCD, if AB is the base and BC is the side, then the perimeter is twice the sum of the lengths of the base and the side.

Text example

Example 1

Let the parallelogram //PQRS have sides PQ = 8 cm, QR = 5 cm, and height = 4 cm. Find the area.

    Area = base × height = PQ × height = 8 cm × 4 cm = 32 cm²

Example 2

In the parallelogram //MNOP, if MN = 12 cm and NO = 7 cm, then find the perimeter.

    Perimeter = 2 × (base + side) = 2 × (12 cm + 7 cm) = 2 × 19 cm = 38 cm

Example 3

What is the length of diagonal OE in the parallelogram //EFGH if EG bisects FH, and FH = 20 cm?

    Since the diagonals bisect each other, GE = GF = 20 cm / 2 = 10 cm

Advanced ideas

1. Properties in coordinate geometry

In coordinate geometry, a parallelogram can also be understood through the coordinates of its vertices. For example, suppose we have a parallelogram with vertices A(x 1, y 1), B(x 2, y 2), C(x 3, y 3), and D(x 4, y 4).

The midpoint formula for every diagonal having the same midpoint can be stated as:

    ((x 1 + x 3) / 2, (y 1 + y 3) / 2) = ((x 2 + x 4) / 2, (y 2 + y 4) / 2)

This equation must be true for the diagonals in a parallelogram to bisect each other.

By understanding these properties, we can solve a myriad of problems related to geometry, physics, and other real-world applications.

Conclusion

Parallelograms play a vital role in geometry. Their properties provide valuable information that becomes extremely useful in solving complex geometric problems. By mastering this fundamental concept, you are laying the foundation for more advanced exploration in the field of mathematics. With enough practice using both conceptual knowledge and real-life applications, handling problems related to parallelograms becomes much more manageable.


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Properties of Parallelograms

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