These notes for CBSE Class 9 Mathematics, Chapter 8, cover the topic of Quadrilaterals. A quadrilateral is defined as a closed figure with four vertices, four sides, four angles, and two diagonals, where no three points are collinear. The sum of the interior angles of any quadrilateral is always 360°. The notes detail various types of quadrilaterals: Trapezium (one pair of opposite sides parallel), Parallelogram (both pairs of opposite sides parallel), Rectangle (all angles 90°), Rhombus (all sides equal), Square (all sides equal and all angles 90°), and Kite (two pairs of adjacent sides equal). It clarifies relationships between these types, noting that squares, rectangles, and rhombuses are parallelograms, while kites and trapeziums are not. Key properties of parallelograms are explained, including that a diagonal divides it into congruent triangles, opposite sides and angles are equal, and diagonals bisect each other. Theorems related to these properties are stated. Finally, the Midpoint Theorem (Basic Proportionality Theorem) is presented with its two statements concerning line segments joining midpoints of triangle sides. These notes are ideal for quick revision before exams.
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A quadrilateral is a closed figure obtained by joining four point (with no three points collinear)
in an order.
Every quadrilateral has :
(i) Four vertices
(ii) Four sides
(iii) Four angles
(iv) Two
diagonals
Statement: The sum of the angles ofa quadrilateral is 360°
1. Trapezium : It is quadrilateral in which one pair of opposite sides are parallel.
2. Parallelogram : It is a quadrilateral in which both the pairs of opposite sides are parallel.
3. Rectangle : It is a quadrilateral whose each angle is 90°. ABCD is a rectangle.
(i) ?A+ ?B = 90° + 90° = 180° ? AD || BC
(ii) ?B+ ?C= 900 + 900 = 180° ? AB || DC
Rectangle ABCD is a parallelogram also.
4. Rhombus : It is a quadrilateral whose all the sides are equal.
5. Square : It is a quadrilateral whose all the sides are equal and each angle is 90°.
6. Kite : It is a quadrilateral in which two pairs of adjacent sides are equal.
Note :
• Square, rectangle and rhombus are all parallelograms.
•Kite and trapezium are not parallelograms.
• A square is a rectangle.
• A square is a rhombus
• A parallelogram is a trapezium.
A parallelogram is a quadrilateral in which opposite sides are parallel. It is denoted by
1. A diagonal of a parallelogram divides it into two congruent triangles.
2. The opposite sides of a parallelogram are equal.
Theorem : If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
3. The opposite angles of a parallelogram are equal.
Theorem : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
4. The diagonals of a parallelogram bisect each other.
Theorem : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
PROPORTIONALITY THEOREM)
Statement 1:
The line segment joining the mid-points of any two sides of a triangle is parallel to the third
side
Statement 2:
The line drawn through the mid-point of one side of a triangle, parallel to another side bisects
the third side.
A quadrilateral is a closed figure formed by joining four points (no three collinear) in order. It has four vertices, four sides, four angles, and two diagonals. The sum of the interior angles of any quadrilateral is always 360°. Opposite sides are parallel and equal, opposite angles are equal, and diagonals bisect each other. A diagonal divides it into two congruent triangles. A rectangle is a quadrilateral where each angle measures 90°. It is also a parallelogram. A square is a quadrilateral with all sides equal and all angles equal to 90°. It is a special type of rectangle and rhombus. The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and half its length. Also, a line through the midpoint of one side, parallel to another, bisects the third side. Yes, a square is a parallelogram because both pairs of its opposite sides are parallel.
Frequently asked questions
What is a quadrilateral?
What is the sum of angles in a quadrilateral?
What are the properties of a parallelogram?
What is a rectangle?
What is a square?
What does the Midpoint Theorem state?
Is a square a parallelogram?
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