Prove that 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. ABC is an isosceles triangle in which AB=AC. AD bisects exterior angle PAC and CD||AB. Show that (i) angle DAC=angle BCA and(ii) ABCD is a parallelogram (||gm).
13. Show that the bisectors of the angles of a parallelogram form a rectangle.
14. 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]
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 a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP=BQ. Show that
(I) tri APB cong Tri CQB
(iv) tri AQB cong Tri CPD
(v) APCQ is a parallelogram.
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.
25. Parallelograms on the same base and between the same parallels are equal in area.