ncert exempler class lX quadrilateral

. • Sum of the angles of a quadrilateral is 360º, 

• A diagonal of a parallelogram divides it into two congruent triangles,

 • In a parallelogram (i) opposite angles are equal (ii) opposite sides are equal (iii) diagonals bisect each other. 

• A quadrilateral is a parallelogram, if (i) its opposite angles are equal (ii) its opposite sides are equal (iii) its diagonals bisect each other (iv) a pair of opposite sides is equal and parallel. 

• Diagonals of a rectangle bisect each other and are equal and vice-versa • Diagonals of a rhombus bisect each other at right angles and vice-versa 

• Diagonals of a square bisect each other at right angles and are equal and vice-versa 

• 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

• A line drawn through the mid-point of a side of a triangle parallel to another side bisects the third side, 

• The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, taken in order, is a parallelogram.

1.The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60º. Find the angles of the parallelogram.

2. E and F are points on diagonal AC of a parallelogram ABCD such that AE = CF. Show that BFDE is a parallelogram

3. E is the mid-point of the side AD of the trapezium ABCD with AB || DC. A line through E drawn parallel to AB intersect BC at F. Show that F is the mid-point of BC.

4. D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. Show that ∆ DEF is also an equilateral triangle

5. Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP = CQ (Fig. ). Show that AC and PQ bisect each other.

6. Show that the quadrilateral formed by joining the mid-points the sides of a rhombus, taken in order, form a rectangle.

7. A square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

8. P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. Prove that PQRS is a rectangle.

9. P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.

10. AB || DE, AB = DE, AC || DF and AC = DF. Prove that BC || EF and BC = EF

   

11. P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram. 

12. ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that ∠A = ∠B and ∠C = ∠D.

13. P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA = AR and CQ = QR.

14. D, E and F are respectively the mid-points of the sides AB, BC and CA of a ΔABC. Prove that by joining these mid-points D, E and F, the ΔABC is divided into four congruent triangles.

15. E and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Prove that EF || AB and EF = ½ (AB + CD).