Question

In the given figure, \[\text{AB}\parallel \text{CD}\] and \[\text{EF}\] is a transversal. If \[\angle \text{AEF = 65}{}^\circ \], \[\angle \text{DFG = 30}{}^\circ \], \[\angle \text{EGF = 90}{}^\circ \] and \[\angle \text{GEF = }x{}^\circ \], find the value of \[x\].

Answer 1

Hint: In this question, it is given that, \[\text{AB}\parallel \text{CD}\] and \[\text{EF}\] is a transversal. Before jumping in the question first we have to understand what a transversal means. A transversal is a line which intersects two or more than two lines, which can or cannot be parallel. When a transversal intersect other lines it forms two types of angles which are:
(i)Alternate angles
(ii)Corresponding angles
In addition to this, the angle sum property of a triangle states that the sum of all angles of a triangle is always equal to $180{}^\circ $.

Complete step-by-step answer:
Given that \[\text{AB}\parallel \text{CD}\] and \[\text{EF}\] is transversal. Then,
\[\Rightarrow \angle \text{EFD = }\angle \text{AEF = 65}{}^\circ \] (Alternate interior angles)
Now, we can find the measure of \[\angle \text{EFG}\] as,
$\begin{align}
  & \Rightarrow \angle \text{EFD = }\angle \text{EFG + }\angle \text{GFD} \\
 & \text{We know that, }\angle \text{EFD = 65}{}^\circ \text{ and }\angle \text{GFD}=\text{30}{}^\circ \\
 & \Rightarrow \text{65}{}^\circ \text{= }\angle \text{EFG + 30}{}^\circ \\
 & \Rightarrow \angle \text{EFG = 65}{}^\circ - 30{}^\circ \\
 & \Rightarrow \angle \text{EFG = 35}{}^\circ \\
\end{align}$
Hence, $\angle \text{EFG = 35}{}^\circ $
Now, in \[\Delta \text{EFG}\]
We know that the angle sum property of a triangle states that the sum of all angles of a triangle is always equal to $180{}^\circ $.
Thus, by using angle sum property of a triangle we get,
 \[\begin{align}
  & \Rightarrow \angle \text{FEG + }\angle \text{EFG + }\angle \text{EGF = 180}{}^\circ \\
 & \text{Now, given that }\angle \text{EGF}=90{}^\circ \text{ and }\angle \text{EFG = }35{}^\circ \text{as mentioned above}\text{.} \\
 & \text{Thus,} \\
 & \Rightarrow x{}^\circ +35{}^\circ +90{}^\circ =180{}^\circ \\
 & \Rightarrow x{}^\circ +125{}^\circ =180{}^\circ \\
 & \Rightarrow x{}^\circ =180{}^\circ -125{}^\circ \\
 & \Rightarrow x{}^\circ =55{}^\circ \\
\end{align}\]
Therefore, the value of \[x{}^\circ \] is equal to \[55{}^\circ \].

Additional Information:
As previously mentioned, when a transversal cuts two lines it makes two types of angles which are alternate and corresponding angles. These are as explained below:
Alternate angles: Alternate angles are angles that are opposite positions relative to transversal intersecting two lines. Further, alternate angles can be divided into two types of angles which are, alternate interior angles and alternate exterior angles.
Corresponding angles: When two lines are crossed by another line which is transversal, the angles of the matching corners are called corresponding angles.

Note: The concept of transversal and various other angles (alternate and corresponding angles) formed by it should be known by the students, in order to solve these questions. In addition to this, the knowledge of angle sum property of a triangle is also required.