CBSE Class 9 Answered

In AMC and BMD

AM = BM (M is midpoint of AB)

AMC = BMD (vertically opposite angles)

CM = DM (given)

AMC @ BMD (by SAS congruence rule)

AC = BD (by CPCT)

And ACM = BDM (by CPCT)

(ii) We have ACM = BDM

But ACM and BDM are alternate interior angles

Since alternate angles are equal.

Hence, we can say that DB || AC

DBC + ACB = 180º (co-interior angles)

DBC + 90º = 180º

DBC = 90º

(iii) Now in DBC and ACB

DB = AC (Already proved)

DBC = ACB (each 90)

BC = CB (Common)

DBC ACB (SAS congruence rule)

(iv) We have DBC ACB

AB = DC (by CPCT)

AB = 2 CM

CM =AB