LEARN MATHS WITH ME ! Quadrilaterals ! Exercise 8.1 (VID 1/2) ! Chapter 8 ! Class 9 !

LEARN MATHS WITH ME ! Quadrilaterals ! Exercise 8.1 (VID 1/2) ! Chapter 8 ! Class 9 ! Quadrilaterals, diagonals, Angle Sum Property of a Quadrilateral. the sum of the angles of a quadrilateral is 360°. Types of Quadrilaterals. One pair of opposite sides of quadrilateral is called a trapezium. Parallelograms. Rectangle. Rhombus. Square.  If two pairs of adjacent sides are equal, it is not a parallelogram, it is called a kite. A square is a rectangle and also a rhombus. A parallelogram is a trapezium. A kite is not a parallelogram. A trapezium is not a parallelogram (as only one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram). A rectangle or a rhombus is not a square. Properties of a Parallelogram.  Another Condition for a Quadrilateral to be a Parallelogram.  The Mid-point Theorem. Theorem 8.1 : A diagonal of a parallelogram divides it into two congruent triangles. Theorem 8.2 : In a parallelogram, opposite sides are equal. Theorem 8.3 : If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. Theorem 8.4 : In a parallelogram, opposite angles are equal. Theorem 8.5 : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram. Theorem 8.6 : The diagonals of a parallelogram bisect each other. Theorem 8.7 : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Theorem 8.8 : A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. Theorem 8.9 : The line segment joining the mid-points of two sides of a triangle is parallel to the third side. Theorem 8.10 : The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side. Example 1 : Show that each angle of a rectangle is a right angle. Example 2 : Show that the diagonals of a rhombus are perpendicular to each other. Example 3 : ABC is an isosceles triangle in which AB = AC. AD bisects exterior angle PAC and CD || AB (see Fig. 8.14). Show that (i) ∠ DAC = ∠ BCA and (ii) ABCD is a parallelogram. Example 4 : Two parallel lines l and m are intersected by a transversal p (see Fig. 8.15). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle. Example 5 : Show that the bisectors of angles of a parallelogram form a rectangle. Example 6 : ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD (see Fig. 8.18). If AQ intersects DP at S and BQ intersects CP at R, show that: (i) APCQ is a parallelogram. (ii) DPBQ is a parallelogram. (iii) PSQR is a parallelogram. Example 7 : In ∆ ABC, D, E and F are respectively the mid-points of sides AB, BC and CA (see Fig. 8.27). Show that ∆ ABC is divided into four congruent triangles by joining D, E and F. Example 8 : l, m and n are three parallel lines intersected by transversals p and q such that l, m and n cut off equal intercepts AB and BC on p (see Fig. 8.28). Show that l, m and n cut off equal intercepts DE and EF on q also. Summary In this chapter, you have studied the following points : 1. Sum of the angles of a quadrilateral is 360°. 2. A diagonal of a parallelogram divides it into two congruent triangles. 3. In a parallelogram, (i) opposite sides are equal (ii) opposite angles are equal (iii) diagonals bisect each other 4. A quadrilateral is a parallelogram, if (i) opposite sides are equal or (ii) opposite angles are equal or (iii) diagonals bisect each other or (iv)a pair of opposite sides is equal and parallel 5. Diagonals of a rectangle bisect each other and are equal and vice-versa. 6. Diagonals of a rhombus bisect each other at right angles and vice-versa. 7. Diagonals of a square bisect each other at right angles and are equal, and vice-versa. 8. The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it. 9. A line through the mid-point of a side of a triangle parallel to another side bisects the third side. 10. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram. quadrilateral,types of quadrilateral,quadrilateral class 9,class 9 maths chapter 8,class 9 chapter 8,quadrilateral shapes,quadrilateral angles,class 9 maths quadrilaterals,class 9,mathematics,class 9 maths,ncert solution,ncert maths class 9,mathematics class 9,tutorial,ncert class 9 quardilateral,class 9 quadrilateral full chapter,quadrilateral class 9 ncert,learn maths with me,learn maths,vikas saini,maths 2020,class 9 2020,exercise 8.1 (vid 1/2)