Quadrilaterals

Exercise 1

1.   The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all angles of the quadrilateral.

2.   If the diagonals of a parallelogram are equal, show that it is a rectangle.

3.   Show that is diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

4.   Show that the diagonals of a square are equal and bisect each other at right angles.

5.   Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

6.   Diagonal AC of a parallelogram ABCD bisects ∠ A (See figure).

Show that:

(i) It bisects ∠ C also.

(ii) ABCD is a rhombus.

7.    ABCD is a rhombus. Show that the diagonal AC bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.

8.   ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C. Show that:

(i) ABCD is a square.

(ii) Diagonal BD bisects both ∠ B as well as ∠ D.

9.    In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (See figure).

Show that:

(i)  Δ APD ≅ Δ CQB

(ii) AP = CQ

(iii) Δ AQB ≅ Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram.


10.  ABCD is a parallelogram and AP and CQ are the perpendiculars from vertices A and C on its diagonal BD (See figure).

Show that:

(i) Δ APB ≅ Δ CQD

(ii) AP = CQ

11.  An Δ ABC and Δ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (See figure).

Show that:

(i) Quadrilateral ABED is a parallelogram.

(ii) Quadrilateral BEFC is a parallelogram.

(iii) AD || CF and AD = CF

(iv) Quadrilateral ACFD is a parallelogram.

(v) AC = DF

(vi) Δ ABC ≅ Δ DEF

12.  ABCD is a trapezium in which AB || CD and AD = BC (See figure).

Show that:

(i) ∠ A = ∠ B

(ii) ∠ C = ∠ D

(iii) Δ ABC ≅ Δ BAD

(iv) Diagonal AC = Diagonal BD.


Exercise 2

13.  ABCD is a quadrilateral in which P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively (See figure). AC is a diagonal. Show that:

(i) 

(ii) PQ = SR

(iii) PQRS is a parallelogram.

14.  ABCD is a rhombus and P, Q, R, S are mid-points of AB, BC, CD and DA respectively. Prove that quadrilateral PQRS is a rectangle.

15.  ABCD is a rectangle and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

16.  ABCD is a trapezium, in which AB 11 DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E, parallel to AB intersecting BC at F (See figure). Show that F is the mid-point of BC.


17.  In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (See figure). Show that the line segments AF and EC trisect the diagonal BD.

18.  Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisect each other.

19.  ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D.