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## Quadrilaterals ☰

A quadrilateral is a 2-dimensional geometric shape that has four sides and four angles. Quadrilaterals are one of the most commonly studied shapes in geometry, and they are used in a wide range of applications, including architecture, engineering, and design.

There are many different types of quadrilaterals, each with its own set of properties and characteristics. Here are some of the most common types of quadrilaterals:

**Square:** A square is a type of quadrilateral that has four equal sides and four right angles. A square is also a type of rectangle and a type of rhombus. **Rectangle:** A rectangle is a type of quadrilateral that has four right angles. The opposite sides of a rectangle are equal in length, but the other two sides may or may not be equal. **Rhombus:** A rhombus is a type of quadrilateral that has four equal sides. The opposite angles of a rhombus are equal, but the other two angles may or may not be equal. **Parallelogram:** A parallelogram is a type of quadrilateral that has opposite sides that are parallel. The opposite sides of a parallelogram are also equal in length. **Trapezium:** A trapezium, also known as a trapezoid, is a type of quadrilateral that has one pair of parallel sides. The other two sides are not parallel and may or may not be equal in length.

Each of these types of quadrilaterals has its own set of properties and characteristics that make them unique. For example, squares have all of the properties of both rectangles and rhombuses, while parallelograms have properties related to parallel lines and vectors.

In addition to these basic types of quadrilaterals, there are also more complex quadrilaterals, such as **kites** and **tangential quadrilaterals**, that have specific geometric properties related to their angles and sides.

Overall, the study of quadrilaterals is an important part of geometry and has a wide range of practical applications in fields such as architecture, engineering, and design. Understanding the properties and characteristics of quadrilaterals can help us to analyze and design complex structures and shapes.

## The interior and exterior angles of a quadrilaterals. ☰

The interior and exterior angles of a quadrilateral are important properties that are used to describe the angles within and outside of the shape.

**Interior Angles of a Quadrilateral:**

The interior angles of a quadrilateral are the angles inside the shape formed by the intersection of its four sides. To find the sum of the interior angles of a quadrilateral, we can use the formula:

sum of interior angles:

\((n-2)\cdot 180^\circ \), where \(n\) is the number of sides in the shape.

For a quadrilateral, \(n=4\). so the sum of its interior angles can be found using:

\((4-2)\cdot 180^\circ =2\cdot 180^\circ =360^\circ \).

This means that the sum of the interior angles of a quadrilateral is always 360 degrees.

**Exterior Angles of a Quadrilateral:**

An exterior angle of a quadrilateral is an angle formed at any vertex of the quadrilateral that is not adjacent to the angle in question. Since a quadrilateral has four vertices, you can draw two exterior angles at each vertex. These angles are congruent to their corresponding interior angles, so when we refer to the exterior angles of a quadrilateral, we take only one of these angles from each vertex.

The sum of the exterior angles of a quadrilateral is 360 degrees.

## Parallelogram. ☰

A parallelogram is a type of quadrilateral with two pairs of opposite parallel sides. Parallelograms are widely studied in geometry and are used in a variety of applications, including architecture, engineering, and design.

**Properties of Parallelograms:**

**Opposite sides are parallel:** In a parallelogram, opposite sides are parallel, meaning they never meet. This property makes the shape useful in applications that require parallel lines. **Opposite sides are equal in length:** In a parallelogram, opposite sides are also equal in length. This property means that a parallelogram is a balanced shape that can be used in designs where symmetry is important. **Opposite angles are equal:** In a parallelogram, opposite angles are also equal. This means that if you know the measure of one angle, you can find the measure of the opposite angle. **Diagonals bisect each other:** The diagonals of a parallelogram bisect each other. This means that the diagonals cut each other in half, and the point where they intersect is the midpoint of both diagonals. **Area:** The area of a parallelogram can be found by multiplying the length of one of its sides by the height of the parallelogram. The height is the perpendicular distance between two parallel sides. **Sum of interior angles:** The sum of the interior angles of a parallelogram is equal to 360 degrees.

**Note:** - A quadrilateral with opposite sides that are congruent and parallel is a parallelogram.
- A quadrilateral with congruent opposite sides is a parallelogram.
- If the diagonals of a quadrilateral intersect and intersect at a point that bisects them, then it is a parallelogram.

**Types of Parallelograms:** **Rectangle:** A rectangle is a type of parallelogram with four right angles. This means that all four angles in a rectangle are equal to 90 degrees. **Rhombus:** A rhombus is a type of parallelogram with four equal sides. This means that all four sides of a rhombus are of the same length. **Square:** A square is a type of parallelogram with four equal sides and four right angles. This means that a square is both a rectangle and a rhombus.

**Applications of Parallelograms:** Parallelograms are used in many fields, including architecture, engineering, and design. They are commonly used in construction to create supports and frameworks for buildings, bridges, and other structures. They are also used in the design of objects and products that require symmetry, such as jewelry, furniture, and packaging.

In summary, parallelograms are a widely studied and versatile shape in geometry. They have a range of properties and characteristics that make them useful in a variety of applications, and understanding their properties and types can help us to analyze and design complex structures and shapes.

## Trapezium. ☰

A trapezium is a quadrilateral in which one pair of opposite sides is parallel.

**Properties of Trapeziums:**

**Parallel sides:** A trapezium has at least one pair of parallel sides, meaning that the sides never meet. **Non-parallel sides:** The remaining two sides of a trapezium are not parallel and may be of different lengths. **Height:** The height of a trapezium is the perpendicular distance between the two parallel sides. **Area:** The area of a trapezium can be found by multiplying the sum of the lengths of the parallel sides by the height of the trapezium, and then dividing by 2 **Angles:** The sum of the interior angles of a trapezium is 360 degrees.

**Types of Trapeziums:** **Isosceles trapezium:** An isosceles trapezium is a trapezium with equal angles and sides that are parallel. **Right trapezium:** A right trapezium is a trapezium with one right angle. **Scalene trapezium:** A scalene trapezium is a trapezium with no equal angles or sides.

**The median line of a trapezoid ** is a line segment that connects the midpoints of the two non-parallel sides of the trapezoid. A trapezoid is a quadrilateral with one pair of parallel sides, so the median line is the line segment that joins the midpoint of the two non-parallel sides. The median line is also known as the mid-segment of a trapezoid.

**Properties of the Median Line of a Trapezoid:**

**Length:** The length of the median line of a trapezoid is equal to the average of the lengths of the two parallel sides. **Parallel:** The median line of a trapezoid is parallel to both parallel sides of the trapezoid. **Midpoints:** The median line of a trapezoid connects the midpoints of the two non-parallel sides. **Bisector:** The median line of a trapezoid bisects the area of the trapezoid. **Relationship with Diagonals:** The median line of a trapezoid is also half the sum of the lengths of the two diagonals of the trapezoid.

## The median line of a triangle. ☰

The median line of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. A triangle is a three-sided polygon, and it has three median lines, one from each vertex. The median lines intersect at a single point, known as the centroid of the triangle. The centroid is the center of gravity of the triangle, and it is equidistant from the three vertices.

**Properties of the Median Line of a Triangle:** **Length:** The length of the median line of a triangle is half the length of the side to which it connects and bisects. **Intersection:** The three median lines of a triangle intersect at a single point, known as the centroid. **Centroid:** The centroid of a triangle is the center of gravity of the triangle and is equidistant from the three vertices. **Area:** The area of a triangle is equal to one-third of the product of the length of any median line and the length of the side to which it connects. **Parallelism:** The median lines of a triangle are parallel to the opposite side of the triangle.