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

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

includes all the questions provided in NCERT Books prepared by Mathematics expert teachers as per CBSE NCERT guidelines from To download our free pdf of Chapter 9 Areas of Parallelograms and Triangles Maths NCERT Solutions for Class 9 to help you to score more marks in your board exams and as well as competitive exams.

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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.


 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

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