ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that

(i) Δ APB ≅Δ CQD

(ii) AP = CQ

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

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:

(i) Δ APD ≅Δ CQB

(ii) AP = CQ

(iii) Δ AQB ≅Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

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 (see Fig. 8.22). 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.

ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that

(i) ∠ A = ∠ B

(ii) ∠ C = ∠ D

(iii) Δ ABC ≅Δ BAD

(iv) Diagonal AC = diagonal BD

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