Question

AB is a line segment and C is the midpoint, D and E are points on the same side of AB such that $\angle DCA=\angle ECB$ and $\angle DBC=\angle EAC$ then (i) $\Delta DBC\cong \Delta EAC$ (ii) $DC=EC$

both statements are

• A. True
• B. False

Answer 1

The correct option is A True

$\Rightarrow$ $AC=BC$ [ $C$ is mid-point of $AB$ ]

$\Rightarrow$ $\angle DCA=\angle ECB$ [ Given ]

$\Rightarrow$ $\angle DBC=\angle EAC$ [ Given ]

Adding $\angle ECD$ to both sides,

$\Rightarrow$ $\angle DCA+\angle ECD=\angle ECB+\angle ECD$

$\Rightarrow$ $\angle ECA=\angle DCB$ ----- ( 1 )

In $△DBC$ and $△EAC$

$\Rightarrow$ $\angle DCB=\angle ECA$ [ From ( 1 ) ]

$\Rightarrow$ $AC=BC$ [ Given ]

$\Rightarrow$ $\angle DBC=\angle EAC$ [ Given ]

$\therefore$ $△DBC\cong △EAC$ [ By $ASA$ congruence test ]

$\Rightarrow$ $DC=EC$ [ By C.P.C.T ]