In the given figure, C is the midpoint of AB. If ∠DCA=∠ECB and ∠DBC=∠EAC, prove that DC=EC.

Question

In the given figure, C is the midpoint of AB. If &#x2220; DCA= &#x2220; ECB and &#x2220; DBC= &#x2220; EAC, prove that DC=EC.

Philoid · Accepted Answer

It is given that AC = BC , ∠DCA = ∠ECB and ∠DBC = ∠EAC.

Adding angle ∠ECD both sides in ∠DCA = ∠ECB, we get,

∠DCA + ∠ECD = ∠ECB + ∠ECD

∴∠ECA = ∠DCB …addition property

Now in ΔDBC and ΔEAC,

∠ECA = ∠DCB

BC = AC

∠DBC = ∠EAC

Hence by ASA postulate, we conclude,

ΔDBC ≅ ΔEAC

Hence, by cpct, we get,

DC = EC