Question

If two parallel lines are intersected by a transversal, then prove that bisectors of any two corresponding angles are parallel.

Answer 1

Hint: First use the property of parallel lines to use the corresponding angles then multiply by $\dfrac{1}{2}$ to both the sides of the equation and from that prove what is given in the question.

Complete step-by-step answer:
In the question we are given two parallel lines intersected by a transversal so we will represent in the figure,

Let AB and CD be two parallel lines and transversal EF intersects then at G and H respectively.
We draw GM and HN such that they bisect $\Delta $EGB and GHD respectively.
There $\Delta $EGB and GHD are corresponding angles.
Now we have to prove that \[GM||HN\] means GM is parallel to HN.
Now as we know that \[AB||CD\] then we can say that \[\angle EGB\text{ }=\angle GHD\] as corresponding angles are equal. So, by multiplying by $\dfrac{1}{2}$ to both sides, we get
$\dfrac{1}{2}\angle EGB=\dfrac{1}{2}\angle GHD$
Now as we took $\angle EGM=\angle 1,\angle GHN=\angle 2$
So we can say that $GM||HN$
Which means GM is parallel to HN as corresponding angles are equal.
Hence it is proved.
Note: Students should know all the properties of parallel lines when cut by a transversal. They should know what are corresponding angles and etc.
Another approach is considering the two parallel lines AB and CD traversed by line EF, as shown in figure below:

Now we know corresponding angles are equal so,
\[\angle EGB\text{ }=\angle GHD\]
From figure, we can rewrite this as,
\[\angle EGM+\angle MGB\text{ }=\angle GHN+\angle NHD\]
Now, it is also given that GM and HN are angle bisector, so above equation can be written as,
\[\begin{align}
  & 2\angle EGM=2\angle GHN \\
 & \Rightarrow \angle EGM=\angle GHN \\
\end{align}\]
Hence, these two angles form corresponding angles of line GM and HN.
Therefore, GM is parallel to HN.
Hence, the bisectors of any two corresponding angles are parallel.