Question

In the given figure, $AB\parallel CD$. Find the value of x.

Answer 1

Hint: At first, draw a line AG at point A parallel to CE. Then, use the fact that $AG\parallel CE$ and alternate angles are equal to find angle GAE. Then, find the angle BAG. After that, finally use the fact that $AB\parallel CD\text{ and }AG\parallel CE$ to tell that angle GAB is equal to angle ECD to get the answer.

Complete step by step answer:

In the question, we are given a figure,

Where $AB\parallel CD$ is given. Now, to proceed we will like to do certain construction such as drawing a line from A such that it is parallel to CE so we can re-draw the figure as,

Hereby construction AG is parallel to CE.
Now, as we know that AG is parallel to CE and AE is transversal then, we can use the property of parallel lines which says alternative angles are equal.
Here, angle AEC and angle GAE are alternative angles then they are equal.
Here, angle AEC is ${{25}^{\circ }}$ So angle GAE is ${{25}^{\circ }}$
Now, we know that angle BAE is ${{105}^{\circ }}$ and GAE is ${{25}^{\circ }}$ so angle BAG will be sum of both angle GAE and BAE which is $\left( {{105}^{\circ }}+{{25}^{\circ }} \right)\Rightarrow {{130}^{\circ }}$
Now, the measure of angle BAG is ${{130}^{\circ }}$. As we know $BA\parallel DC, AG\parallel CE$ so we can say that angle BAG is equal to DCE using the property of corresponding angles.
So, as we know that corresponding angles are equal in parallel lines then we can write,
\[\begin{align}
  & \angle DCE=\angle BAG \\
 & \Rightarrow x={{130}^{\circ }} \\
\end{align}\]
Thus, value of x is ${{130}^{\circ }}$
Note:
 Students can do the same question by using another method. They can do construction C such that CG is parallel to AE and then using properties of corresponding angles and alternate angles to get the desired result.