IX M SAII QUADRILATERALS ASSIGNMENT 1

Prove the followings:

1. A diagonal of a parallelogram divides it into two congruent triangles. 
2. In a parallelogram, opposite sides and angle are equal. 
3. If each pair of opposite sides of quadrilateral is equal, then it is a parallelogram. 
4. If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram. 
5. The diagonals of a parallelogram bisect each other. 
6. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. 
7. A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. 
8. The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side. 
9. The line segment joining the mid- points of the two sides of a triangle is parallel to the third side. 
10. Show that each angle of a rectangle is a right angle. 
11. Show that the diagonal of a rhombus are perpendicular to each other. 
12. Show that the bisectors of the angles of a parallelogram form a rectangle. 
13. ABCD is a parallelogram (||gm) in which P and Q are mid-points of opposite side AB and CD. If AQ intersects DP at S and BQ intersects CO at R, show that

(i) APCQ is ||gm                              (ii)DPBQ is ||gm                             (iii) PSQR is ||gm

14. In Triangle ABC, D, E and Fare respectively the mid points of sides AB, BC and CA. Show that triangle ABC is divided into four congruent triangle by joining D, E and F
15. If the diagonal of a parallelogram are equal, then show that it is a rectangle. 
16. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus. 
17. Show that the diagonals of a square are equal and bisect each other at right angles. 
18. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. 
19. In 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. Show that:

(i) Quadrilateral ABCD is a parallelogram.

(ii) Quadrilateral BEFC is a parallelogram.

(iii) AD||CF and AD=CF

(iv) Quadrilaterals ACFD is a parallelogram

(v) AC=DF

(vi) Δ ABC ≅ Δ DEF. 

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

(i) SR||AC and SR =1/2 AC                          (ii) PQ=SR                          (iii) PQRS is a parallelogram. 

21. ABCD is a rhombus and P, Q, R and S are the mid- point of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle. 
22. ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus. 
23. Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other. 
24. ABC is a triangle right angle at C. A line through the mid-points M of hypotenuse AM and parallel to BC intersects AC at D. Show that

(i) D is the mid –point of AC                      (ii) MD ┴ AC                      (iii) CM=MA=1/2 AB.