Question

In the following figure \[t\] is the transversal of \[l\] and \[m\]. If \[\angle {\rm{APQ}} = {\rm{72}}^\circ \] and \[\angle {\rm{DQF}} = 108^\circ \]. Prove that \[l\parallel m\].

Answer 1

Hint:
Here we will use the concept of the corresponding angles to prove that the line \[l\] is parallel to the line \[m\]. First we will use the concept of straight line to find the angle \[\angle {\rm{BQF}}\]. Then we will compare the angle \[\angle {\rm{BQF}}\]with the angle \[\angle {\rm{APQ}}\]. If the value of both the angles is the same then with the concept of the corresponding angles we can say the line \[l\] is parallel to the line \[m\].

Complete step by step solution:
Firstly we will use the concept of straight line. As on the straight line, the sum of all the angles on one side of the line makes an angle of \[180^\circ \]. Therefore, considering the line \[m\] as straight line we get
\[\angle {\rm{BQF}} + \angle {\rm{DQF}} = 180^\circ \]
By putting the value of angle \[\angle {\rm{DQF}} = 108^\circ \] in the equation we will get the value of the angle \[\angle {\rm{BQF}}\], we get
\[ \Rightarrow \angle {\rm{BQF + }}108^\circ = 180^\circ \]
\[ \Rightarrow \angle {\rm{BQF}} = 180^\circ - 108^\circ = {\rm{72}}^\circ \]
We can clearly see that the angle \[\angle {\rm{BQF}}\] is equal to the angle\[\angle {\rm{APQ}}\]. So according to the concept of corresponding angles, if the corresponding angles between two lines are equal then the two lines are said to be parallel to each other.

Therefore, the line \[l\] is parallel to the line \[m\].
Hence proved.

Note:
Here we should not confuse the corresponding angles with the vertically opposite angles. We have to note that corresponding angles are the angles present on the corresponding corners. If the corresponding angles between two lines are equal then the two lines are said to be parallel to each other. Vertically opposite angles or vertical angles those angles that are opposite to each other when two lines intersect each other. A pair of vertically opposite angles (vertical angles) is always equal to each other. Transversal line is the line which intersects or crosses other two lines. We should not confuse the corresponding angles with the vertically opposite angles.