Areas of Parallelogram & Triangles / Quadrilaterals
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OUADRILATERALS Sub'ect : Maths Class : IX KEY POINTS : Board : CBSE Definition : A plane figure bounded by FOUR line segments AB, BC, CD and DA is called a Quadrilateral and is written as quad ABCD or ABCD. co s C u ve OPPO s / Co A/ s ECU SIDES OPP0S/ S/ DES Theorem 1 : The sum of the four angles of a quadrilateral is 3600 TYPES OF QUADRILATERALS TRAPEZIUM : A quadrilateral having exactly ONE PAIR OF PARALLEL LINES is called a Trapezium. ISOSCELES TRAPEZIUM : A trapezium is said to be an isosceles trapezium if its non-parallel sides are equal. PARALLELOGRAM : A quadrilateral is a parallelogram if it' s both pair of opposite sides are parallel. RHOMBUS : A parallelogram having all sides equal is called Rhombus.
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RECTANGLE : A parallelogram whose each angle is right angle is called a Rectangle. SOUARE : A parallelogram having all sides equal and each angle equal to aright angle is called a Square. KITE : A quadrilateral is a kite if it has two pair of equal adjacent sides and unequal opposite sides. PROPERTIES OF A PARALLELOGRAM Theorem 1 : A diagonal of a parallelogram divides it into two congruent triangles. Theorem 2 : In a parallelogram opposite sides are equal. Theorem 3 : The opposite angles of a parallelogram are equal. Theorem 4 : The diagonals of a parallelogram BISECT each other. Theorem 5 : In a parallelogram the BISECTORS of any two consecutive angles intersect at right angles. Theorem 6 : If diagonal of a parallelogram bisects one of the angle of the parallelogram, it also bisects the second angle. Also it is a RHOMBUS. Theorem 7 : The angle bisectors of a parallelogram form a RECTANGLE. SUFFICIENT CONDITIONS FOR A OUADRILATERAL TO BE A PARALLELOGRAM A quadrilateral is a Parallelogram 1. 2. 3. 4. If both pair of opposite sides are equal. If both pair of opposite angles are equal. If the diagonals bisect each other. If a pair of opposite sides are parallel as well as equal. PROPERTIES OF A RECTANGLE, RHOMBUS AND A SQUARE Theorem 1 : Each of the four angles of a Rectangle is a right angle.
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Theorem 2 : Each of the four sides of a Rhombus is of the same length. Theorem 3 : Each of the angles of a square is a right angle and each of the four sides is of the same length. Theorem 4 : The diagonals of a rectangle are of equal length. Theorem 5 : If two diagonals of a parallelogram are equal, it is a rectangle. Theorem 6 : The diagonals of a rhombus are perpendicular to each other. Theorem 7 : If the diagonals of a parallelogram are perpendicular , then it is a rhombus. Theorem 8 : The diagonals of a square are equal and perpendicular to each other. Theorem 9 : If the diagonals of a parallelogram are equal and intersect at right angles, then the parallelogram is a Square SOME USEFUL FACTS ABOUT A TRIANGLE Theorem 1 ( MID POINT THEOREM ) : The line segment joining the MID PONTS of any two sides of a triangle is parallel to the third side and equal to half of it. Theorem 2 : The line drawn through the mid-point of one side of a triangle, parallel to another side , intersects the third side at its mid-point. Theorem 3 : The quadrilateral formed by joining the mid points of the sides of a Quadrilateral in order is a parallelogram.
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