Question

The following figure has shown a triangle ABC in which AB = AC. M is a point on AB and N is a point on AC such that BM = CN.
Prove that:  (i) AM = AN  (ii) ΔAMC ≅ ΔANB

Answer 1

In ΔABC, AB = AC. m and N are points on
AB and AC such that BM = CN
BN and CM are joined

(i) In ΔAMC and ΔANB
AB = AC               ...[ Given ]  ...(1)
BM = CN              ....[ Given ] ...(2)
Subtracting (2) from (1), we have
AB - BM = AC - CN
⇒ AM = AN                   ...(3)

(ii) Consider the triangles AMC and ANB
AC = AB                      ...[ given ] 
∠AMC =  ∠ANB         ...[ common 90° ]
AM = AN                  ....[ from ( 3 ) ]
∴ By the Side-Angel-Side Criterion of congruence, we have ΔAMC ≅ ΔANB