Question

In the diagram above, $\angle DMQ$ is a right angle, and $\angle BMQ={36}^{\circ }$, then the value of $\angle 1=$

• A. ${54}^{\circ }$
• B. ${36}^{\circ }$
• C. ${64}^{\circ }$

Answer 1

The correct option is A ${54}^{\circ }$

Given $\angle DMQ$ is a right angle.
$\Rightarrow \angle DMQ={90}^{\circ }$
Now, the lines $AB,PQ\&CD$ are intersecting each other at a point $M$
$\Rightarrow \angle 1=\angle BMC$
Now, $\angle DMQ,\angle BMQ,\angle BMC$ are all lying on a stright line.
$\Rightarrow \angle DMQ+\angle BMQ+\angle BMC={180}^{\circ }$
$\Rightarrow {90}^{\circ }+{36}^{\circ }+\angle BMC={180}^{\circ }$
$\Rightarrow \angle BMC={180}^{\circ }-({90}^{\circ }+{36}^{\circ })={54}^{\circ }$
$\therefore \angle BMC={36}^{\circ }$
Now, $\angle BMC$ and $\angle AMD$ are vertical angles.
$\Rightarrow \angle AMD=\angle BMC={54}^{\circ }$