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Mathematics · Class 9 · Chapter 10: Parallelograms and Triangles

8 min

Key points

A parallelogram is a quadrilateral whose opposite sides are parallel
Opposite sides of a parallelogram are equal and parallel
Opposite angles of a parallelogram are equal
Diagonals of a parallelogram bisect each other
A diagonal divides a parallelogram into two congruent triangles
Triangles on the same base and between the same parallels are equal in area
A triangle and a parallelogram on the same base and between the same parallels have the ratio of areas 1:2
The median of a triangle divides it into two triangles of equal area

Definitions

Parallelogram: A quadrilateral whose opposite sides are parallel is called a parallelogram
Rectangle: A parallelogram whose all angles are right angles is called a rectangle
Rhombus: A parallelogram whose all sides are equal is called a rhombus
Square: A rectangle whose all sides are equal is called a square
Median of Triangle: A line segment joining a vertex of a triangle to the midpoint of the opposite side is called median

Worked example

Prove that the diagonals of a parallelogram bisect each other. (5 marks)

Complete geometric proof showing triangles AOB and COD are congruent, leading to conclusion that O bisects both diagonals

Common mistakes

Students often forget to write 'Given' and 'To Prove' in geometry proofs
Confusing properties of rectangle and rhombus - remember rectangle has equal diagonals, rhombus has perpendicular diagonals
Not labeling vertices correctly in diagrams - always use ABCD for parallelograms
Writing angle measures incorrectly - opposite angles are equal, adjacent angles are supplementary
Forgetting to state which congruence rule (SAS, ASA, SSS) is being used in proofs

Quick recap

Parallelograms have opposite sides equal and parallel, with diagonals bisecting each other. Triangles on same base between same parallels have equal areas. Area relationships are key - parallelogram area equals base × height, triangle area equals ½ × base × height. All proofs must use congruency theorems (SSS, SAS, ASA, RHS) as stated in textbook.

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