(i) In ΔAMC and ΔBMD,
AM = BM (M is the mid-point of AB)
∠AMC = ∠BMD (Vertically opposite angles)
CM = DM (Given)
∴ ΔAMC ≅ ΔBMD (By SAS congruence rule)
∴ AC = BD (By CPCT)
And, ∠ACM = ∠BDM (By CPCT)
(ii) ∠ACM = ∠BDM
However, ∠ACM and ∠BDM are alternate interior angles. Since
alternate angles are equal, It can be said that DB || AC
∠DBC + ∠ACB = 180º (Co-interior angles)
∠DBC + 90º = 180º
∠DBC = 90º
(iii) In ΔDBC and ΔACB,
DB = AC (Already proved)
∠DBC = ∠ACB (Each 90 )
BC = CB (Common)
∴ ΔDBC ≅ ΔACB (SAS congruence rule)
(iv) ΔDBC ≅ ΔACB
AB = DC (By CPCT)
AB = 2 CM
∴ CM =1/2 AB