Question

State Whether the following statement is True or False:
In the figure, $C$ is the mid-point of $AB$, $\angle BAD=\angle CBE,\angle ECA=\angle DCB$ then $DA=EB$.

• A. True
• B. False

Answer 1

The correct option is A True

Given $\angle P=\angle Q$

$⟹$ $\angle ECA=\angle DCB$.

Now adding $\angle X$ both side we get:

$\angle P+\angle X=\angle Q+\angle X$ ........$\left[1\right]$.

Now in triangles $\Delta ACD$ and $\Delta BCE$,

$\angle A=\angle B$........[given]

$⟹$ $\angle BAD=\angle ABE$

$AC=BC$...........[$C$ is midpoint of $AB$]

$\angle ACD=\angle BCE$.......$\left[1\right]$

So, $\Delta ACD\cong \Delta BCE$......[by $ASA$ congruency criterion]

Hence by $CPCT$, the corresponding parts are also equal.

Then, $AD=BE$.

Therefore, the given statement is true.

Hence, option $A$ is correct.