Question

State true or false:

In a pentagon $ABCDE$, $AB=AE,BC=ED$ and $\angle ABC=\angle AED$, then

$\angle BCD=\angle EDC$

• A. True
• B. False

Answer 1

The correct option is A True

In a given pentagon, ABCDE, $AB=AE,BC=EDand\angle ABC=\angle AED$. We join AC and AD to form our diagram as shown above:

Now since in $△ABC$ and $△AED$,

$AB=AE,BC=EDand\angle ABC=\angle AED$ is given, therefore, by SAS rule we get$△ABC\cong △AED$

Thus, by CPCT we get $AC=AD$ and $\angle BCA=\angle EDA$....(eqn 1)

Now in $△CAD$, we know $AC=AD$

Therefore, from base angle theorem, the following angles are equal:

$\angle ACD=\angle ADC$.....(eqn 2)

$\angle BCD=\angle BCA+\angle ACD$.....(eqn 3)

$\angle EDC=\angle EDA+\angle ADC$

So from equations 1 and 2,

$\angle EDC=\angle BCA+\angle ACD$.......(eqn 4)

Thus from equation 3 and 4 we get

$\angle BCD=\angle EDC$.

Hence proved.