Question

In the given figure, if $AE\parallel DC$ and $AB=AC,$ the value of $\angle ABD$ is

(a) ${70}^{\circ }$

(b) ${110}^{\circ }$

(c) ${120}^{\circ }$

(d) ${130}^{\circ }$

Answer 1

The correct option is (b): ${110}^{\circ }$

We have to find the value of $\angle ABD$ in the following figure.

It is given that

$\angle MAE={70}^{\circ }$

$\angle NAC={70}^{\circ }$(Vertically opposite angle)

Now $\angle MAE+\angle EAB+\angle BAC={180}^{\circ }$ (angles on the same side of straight line) …… (1)

Similarly $\angle EAB+\angle BAC+\angle NAC={180}^{\circ }$ (angles on the same side of straight line) …… (2)

From equation (1) we have

$\angle EAB+\angle BAC={180}^{\circ }-{70}^{\circ }={110}^{\circ }$

$\angle EAC=\angle EAB+\angle BAC={110}^{\circ }$

$\angle ACF=\angle EAC$ ( Alternate Interior angle)

Now, $\angle EAC={110}^{\circ }$

So, $\angle ACF={110}^{\circ }$

Now $\angle ACF=\angle ABD$ (exterior angles of same triangle)

Since $\angle ACF=\angle ABD={110}^{\circ }$