Diagonal AC of a parallelogram ABCD bisects ∠A. Show that (i) It bisects ∠C also, (ii) ABCD is a rhombus.

□ABCDis a parallelogram givensideAD≅sideBC...opposite sides of a parallelogram are congruent (1)In△ADCand△ABC,sideAD≅sideBCfrom (1)∠DAC=∠ACB.Alternate anglesideAC≅sideACcommon side△ADC≅△ABC.SAS test of congruence∴∠DCA=∠BCAc.a.c.t. (2)∴DiagACis bisector of∠C.We know, opposite angles of a parallelogram are congruent.∠DAB=∠BCD∴12∠DAB=12∠BCD∴∠BAC=∠BCA(3)In△ABC,∠BAC=∠BCAfrom (3)∴BC=ABConverse of isosceles triangle theoremSimilarly, we can proveAD=DC.Since, the adjacent sides of a parallelogram are equal,□ABCDis a rhombus.