Properties of a Parallelogram

A parallelogram is a quadrilateral (a four-sided polygon) with two pairs of parallel sides. This fundamental geometric shape boasts several key properties that are crucial for understanding geometry. These properties allow us to calculate various attributes like angles, side lengths, and areas.

Formulae

While there isn’t a single “formula” for a parallelogram, we use the following relationships derived from the properties:

• Area: $Area = base \times height$ (where height is the perpendicular distance between the base and the opposite side)
• Perimeter: $Perimeter = 2 \times (side1 + side2)$

Examples

Let’s consider some examples to illustrate the properties:

Example-1: If a parallelogram has two sides of length 5 cm and 8 cm, and the angle between these sides is 60 degrees, then the opposite sides will also have lengths 5 cm and 8 cm. Knowing the angle and the base allows you to calculate the height using trigonometry for area calculation.

Example-2: If the diagonals of a parallelogram intersect at a point, and one segment of a diagonal is 3 cm, the other segment of the same diagonal will also be 3 cm (since diagonals bisect each other). The other diagonal segments will also bisect each other according to the length of the diagonal.

Theorem with Proof

Theorem: Opposite sides of a parallelogram are equal in length.

Proof:

1. Given: Parallelogram ABCD, with AB || CD and AD || BC.
2. Draw a diagonal: Draw diagonal AC.
3. Congruent triangles: Because AB || CD, and AC is a transversal, $\angle BAC \cong \angle DCA$ (alternate interior angles). Similarly, because AD || BC, and AC is a transversal, $\angle BCA \cong \angle DAC$.
4. Common Side: AC is a common side to both triangles ABC and CDA.
5. ASA Congruence: By the Angle-Side-Angle (ASA) congruence postulate, $\triangle ABC \cong \triangle CDA$.
6. Corresponding Parts: Since the triangles are congruent, their corresponding sides are equal. Therefore, AB = CD and AD = BC (corresponding parts of congruent triangles are congruent – CPCTC).
7. Conclusion: Thus, opposite sides of a parallelogram are equal in length.

Common mistakes by students

• Confusing properties: Mixing up the properties of parallelograms with those of other quadrilaterals, like rectangles or rhombuses. Remember that rectangles and rhombuses are special types of parallelograms.
• Incorrectly applying area formula: Using an incorrect height measurement when calculating the area. Always ensure the height is perpendicular to the base.
• Assuming all angles are right angles: Failing to recognize that only rectangles and squares (special types of parallelograms) have right angles.

Real Life Application

Parallelograms are used in numerous real-world applications:

• Architecture and Construction: Parallelograms can be seen in the designs of buildings, bridges, and other structures. They contribute to structural stability and aesthetic appeal.
• Engineering: Parallelograms are fundamental to understanding forces and motions. They are used in the design of linkages and mechanisms.
• Art and Design: Artists and designers utilize the properties of parallelograms to create visually appealing compositions, from paintings to graphic designs.
• Tiling and Paving: Parallelograms (especially rectangles and squares) are used in tiling floors and paving roads, due to their ability to fit together without gaps or overlaps.

Fun Fact

The word “parallelogram” comes from the Greek words “para” (meaning beside or alongside) and “gramma” (meaning line). It perfectly describes the shape’s core characteristic: sides running parallel to each other!

Recommended YouTube Videos for Deeper Understanding

Q.1 The lengths of two adjacent sides of a parallelogram are 8 cm and 12 cm. What is the perimeter of the parallelogram?

Check Solution

Ans: B

The perimeter is twice the sum of adjacent sides: $2 * (8 + 12) = 40$ cm

Q.2 In parallelogram ABCD, angle A is $60^\circ$. What is the measure of angle C?

Check Solution

Ans: B

Opposite angles of a parallelogram are equal.

Q.3 The diagonals of a parallelogram

Check Solution

Ans: C

Diagonals of a parallelogram bisect each other.

Q.4 If the diagonals of a parallelogram are 10 cm and 14 cm, what can you conclude about the triangles formed by the diagonals?

Check Solution

Ans: B

Each diagonal divides the parallelogram into two congruent triangles.

Q.5 In parallelogram PQRS, side PQ = 15 cm and the distance between PQ and RS is 8 cm. What is the area of the parallelogram?

Check Solution

Ans: C

Area = base * height = 15 * 8 = 120 $cm^2$

Next Topic: Conditions for a Quadrilateral to be a Parallelogram