Finding the angle of $\angle x$ and $ \angle y $

Question

In the figure below, $ABCD$ is a rhombus and $ADE$ is a straight line. $\angle DAB = 51°$ and $\angle DCE = 42°$. what is the value of $x$ and $y$?

Answer 1

Since $ABCD$ is a rhombus, $\angle A = \angle DCB$, and $DC=CB$. Thus, $\triangle DCB$ is isosceles, and you can figure out $y$ from that.

Once you find $y$, you should be able to figure out what the degree measure of $\angle ADC$ is, and from there, you can get $\angle EDC$. From there, you can get $x$.

Answer 2

As $AB||CD$, $\angle BAD = \angle EDC ...(i)$. So, $x= \pi -$ (other two angles of $\triangle DCE$ ).

And, as $BD$ is a diagonal of a parallelogram, it bisects $y$, which is equal to a $\angle ADC$. And now, from the same statement as $(i)$, we know, $\angle ADC = \pi -\angle BAD$, which isequal to $y$.
(Here, of course, we may use $180^o$, not $\pi$).