3.3-Kinds of Quadrilaterals

3.3-Kinds of Quadrilaterals Important Formulae

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3.3 - Kinds of Quadrilaterals
  • A quadrilateral is a polygon with four sides.
  • Types of quadrilaterals include:
    • Parallelogram: Opposite sides are parallel and equal.
    • Rectangle: A parallelogram with four right angles.
    • Rhombus: A parallelogram with all sides equal.
    • Square: A rectangle with all sides equal.
    • Trapezium: A quadrilateral with one pair of parallel sides.
    • Kite: Two pairs of adjacent sides are equal.
  • Properties are defined by the relationships between sides, angles, and diagonals.
  • Area formula for a rectangle: $A = l \times b$ where $l$ is length and $b$ is breadth.
  • Area formula for a rhombus: $A = \frac{1}{2} \times d_1 \times d_2$ where $d_1$ and $d_2$ are the diagonals.

3.3 - Kinds of Quadrilaterals

Quadrilaterals are polygons with four sides and four angles. They can be classified into various types based on the properties of their sides, angles, and diagonals. Below are the main kinds of quadrilaterals.

1. Parallelogram

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. The key properties of a parallelogram include:

  • Opposite sides are equal in length.
  • Opposite angles are equal.
  • Diagonals bisect each other.
  • Consecutive angles are supplementary (sum to 180°).

The area of a parallelogram is given by the formula:

$\text{Area} = \text{Base} \times \text{Height}$

2. Rectangle

A rectangle is a special type of parallelogram where each angle is a right angle (90°). It has the following properties:

  • All angles are 90°.
  • Opposite sides are equal and parallel.
  • The diagonals are equal in length.

The area of a rectangle is calculated as:

$\text{Area} = \text{Length} \times \text{Breadth}$

3. Square

A square is a quadrilateral that is both a rectangle and a rhombus. All four sides are equal in length, and all angles are 90°. Its properties include:

  • All sides are equal.
  • All angles are 90°.
  • The diagonals are equal in length and bisect each other at right angles.
  • The diagonals divide the square into four congruent right-angled triangles.

The area of a square is:

$\text{Area} = \text{Side}^2$

4. Rhombus

A rhombus is a parallelogram where all sides are of equal length. Its properties are:

  • All sides are equal in length.
  • Opposite angles are equal.
  • Diagonals bisect each other at right angles and bisect the angles of the rhombus.
  • Consecutive angles are supplementary.

The area of a rhombus can be calculated using the formula:

$\text{Area} = \frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2$

5. Trapezium (or Trapezoid in North America)

A trapezium is a quadrilateral with one pair of parallel sides. The non-parallel sides are called the legs. Its properties include:

  • Only one pair of opposite sides are parallel.
  • The sum of the angles on the same side is 180°.
  • The area of a trapezium is given by:

$\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h$

Where $b_1$ and $b_2$ are the lengths of the parallel sides, and $h$ is the height (the perpendicular distance between the parallel sides).

6. Kite

A kite is a quadrilateral with two pairs of adjacent sides that are equal. The properties of a kite are:

  • Two pairs of adjacent sides are equal.
  • One pair of opposite angles are equal (the angles between the unequal sides).
  • The diagonals intersect at right angles and one diagonal bisects the other.

The area of a kite is calculated by:

$\text{Area} = \frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2$

7. General Quadrilateral

A general quadrilateral does not have any specific properties like the other types mentioned above. Its sides and angles can vary in length and measure, and the diagonals may or may not be equal or bisect each other. The sum of the interior angles of any quadrilateral is always 360°.

To calculate the area of a general quadrilateral, you might need specific information about the lengths of its sides or diagonals, or use methods like triangulation or coordinate geometry.

The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

Solution:

Question:

The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

Solution:

Let the measures of the two adjacent angles be $3x$ and $2x$ respectively.

In a parallelogram, the sum of adjacent angles is always $180^\circ$. Therefore, we can write the equation:

$3x + 2x = 180^\circ$

Simplifying this:

$5x = 180^\circ$

Now, solving for $x$:

$x = \frac{180^\circ}{5} = 36^\circ$

Now, substitute $x = 36^\circ$ into the expressions for the angles:

First angle = $3x = 3 \times 36^\circ = 108^\circ$

Second angle = $2x = 2 \times 36^\circ = 72^\circ$

Thus, the measures of the two adjacent angles are $108^\circ$ and $72^\circ$.

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Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Solution:

Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Let the measure of each of the two adjacent angles be $x$.

In a parallelogram, the sum of adjacent angles is always $180^\circ$. So, we have:

$x + x = 180^\circ$

Therefore, $2x = 180^\circ$

Solving for $x$, we get:

$x = \frac{180^\circ}{2} = 90^\circ$

Thus, each of the two adjacent angles measures $90^\circ$.

Since opposite angles of a parallelogram are equal, the other two angles also measure $90^\circ$.

Hence, the measure of each angle of the parallelogram is $90^\circ$.

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