# 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:

(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.