# NCERT Solutions for  Maths Chapter 9 – Areas of Parallelograms and Triangles PDF

## Free PDF of NCERT Solutions for Class 9 Maths Chapter 9 – Areas of Parallelograms and Triangles

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## Parallelogram

The term ‘parallelogram’ was derived from the Greek word ‘parallelogrammon’ which stands for “bounded by parallel lines”. Hence, a parallelogram is a quadrilateral that is bounded by parallel lines. It is a shape in which the opposite sides are parallel and equal. Parallelograms are classified into three main types: square, rectangle, and rhombus, and each of them has its own unique properties. In this section, we will learn about a parallelogram, how to find the area of a parallelogram and other aspects related to a parallelogram along with the solved examples.

### What is a Parallelogram?

A parallelogram is a special kind of quadrilateral that is formed by parallel lines. The angle between the adjacent sides of a parallelogram may vary but the opposite sides need to be parallel for it to be a parallelogram. A quadrilateral will be a parallelogram if its opposite sides are parallel and congruent. Hence, a parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel and equal.

### Characteristics

simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true

• Two pairs of opposite sides are parallel (by definition).
• Two pairs of opposite sides are equal in length.
• Two pairs of opposite angles are equal in measure.
• The diagonals bisect each other.
• One pair of opposite sides is parallel and equal in length.
• Adjacent angles are supplementary.
• Each diagonal divides the quadrilateral into two congruent triangles.
• The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
• It has rotational symmetry of order 2.
• The sum of the distances from any interior point to the sides is independent of the location of the point. This is an extension of Viviani's theorem.)
• There is a point X in the plane of the quadrilateral with the property that every straight line through X divides the quadrilateral into two regions of equal area.
• Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.

## crossed square

A crossed square is a faceting of the square, a self-intersecting polygon created by removing two opposite edges of a square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih2, order 4. It has the same vertex arrangement as the square, and is vertex-transitive. It appears as two 45-45-90 triangle with a common vertex, but the geometric intersection is not considered a vertex.

• A crossed square is sometimes likened to a bow tie or butterfly. the crossed rectangle is related, as a faceting of the rectangle, both special cases of crossed quadrilaterals.
• The interior of a crossed square can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
• A square and a crossed square have the following properties in common:
• Opposite sides are equal in length.
• The two diagonals are equal in length.
• It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
• It exists in the vertex figure of a uniform star polyhedra, the tetrahemihexahedron.
∆∆ reference by Wikipedia