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

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

3. Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig.). Show that:

(i) it bisects ∠C also,

(ii) ABCD is a rhombus.

4. 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 ∠B as well as ∠D.

5. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig.). Show that:

(i) ΔAPD ≅ ΔCQB

(ii) AP = CQ

(iii) ΔAQB ≅ ΔCPD

(iv) AQ = CP

(v) APCQ is a parallelogram

6. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig.). Show that:

(i) ΔAPB ≅ ΔCQD

(ii) AP = CQ

7. ABCD is a trapezium in which AB || CD and AD = BC (see Fig.). Show that:

(i) ∠A = ∠B

(ii) ∠C = ∠D

(iii) ΔABC ≅ ΔBAD

(iv) diagonal AC = diagonal BD

[Hint :Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]