Question

In figure, \[\angle 1 = {60^ \circ }\] and \[\angle 6 = {120^ \circ }\]. Show that the lines \[m\] and \[n\] are parallel.

Answer 1

Hint:We have to show that the lines \[m\] and \[n\] are parallel. We know that vertically opposite angles are equal and also, if the sum of two interior angles on the same side of a line is \[{180^ \circ }\], then the lines are parallel. We will use this to show that \[m\] and \[n\] are parallel.

Complete step by step answer:
From the figure we have \[\angle 1 = {60^ \circ }\] and \[\angle 6 = {120^ \circ }\].
As we know that vertically opposite angles are equal. Therefore, from the figure we get,
\[ \Rightarrow \angle 1 = \angle 3\]
Therefore, we have
\[ \Rightarrow \angle 3 = \angle 1 = {60^ \circ }\]
Now,
\[ \Rightarrow \angle 3 + \angle 6 = {60^ \circ } + {120^ \circ }\]
On simplification we get,
\[ \Rightarrow \angle 3 + \angle 6 = {180^ \circ }\]
As we know, if the sum of two interior angles on the same side of a line is \[{180^ \circ }\], then the lines are parallel.

Therefore, \[m\] and \[n\] are parallel.

Additional information: If a transversal cut a pair of the lines in such a way that the angles of any pair of interior angles on the same side of transversal are supplementary or the angles of any pair of corresponding angles are equal or the angles of any pair of alternate interiors (or exterior) angles are equal then the lines are parallel.

Note:We can also solve this problem by another method. As we know, a linear pair is defined as two adjacent angles that add up to \[{180^ \circ }\] or two angles which when combined together form a straight angle.
Therefore, from the figure we can write
\[ \Rightarrow \angle 5 + \angle 6 = {180^ \circ }\]
On putting the value of \[\angle 6\], we get
\[ \Rightarrow \angle 5 + {120^ \circ } = {180^ \circ }\]
On simplification, we get
\[ \Rightarrow \angle 5 = {60^ \circ }\]
Also,
 \[ \Rightarrow \angle 1 = {60^ \circ }\]
Therefore, we get
\[ \Rightarrow \angle 5 = \angle 1 = {60^ \circ }\]
But, \[\angle 5\] and \[\angle 1\] are corresponding angles.
As corresponding angles are equal if the transversal intersects two parallel lines.
Therefore, \[m\] and \[n\] are parallel.