How can both angles be equal?

Question

In the given figure $ABCD$ and $EFCD$ are parallelograms on the same base $DC$. My textbook says that $\widehat{ADE}= \widehat{FCB}$. But I failed to understand why(the reason) these two angles are equal? The reason which my textbook presents is DE is parallel to CF and AD is parallel to BC.Hence,angle between DE and AD=angle between CF and BC How can both the angles be equal by the given reason in my textbook?

Answer 1

From the definition of transversal we know that:

$\bullet \hspace{0.5 cm} \angle DAE \cong \angle CBF$

$\bullet \hspace{0.5 cm} \angle AED \cong \angle BFC$

Consider triangles $\Delta ADE, \Delta BCF$ and the fact that the sum of angles in a triangle is $180$:

$m\angle DAE+m\angle AED + m\angle ADE = 180$ and $m\angle CBF+m\angle BFC + m\angle FCB = 180$

Thus:

$m\angle DAE+m\angle AED + m\angle ADE = m\angle CBF+m\angle BFC + m\angle FCB $

Substituting values we obtain:

$m\angle CBF+m\angle BFC + m\angle ADE = m\angle CBF+m\angle BFC + m\angle FCB $

Canceling out we are left with:

$m\angle ADE = m\angle FCB $

Answer 2

We need to subtract two angles: