Grade IX - Mathematics

Author:
Rahul Chandra Saha
M.Sc., Mathematics

Lines And Angles

Transversal: Whenever a line intersects two or more lines at different points, the line is called a Transversal.

In the images you see line $\overleftrightarrow{EF}$ is a transversal to the lines $\overleftrightarrow{AB}$ and  \overleftrightarrow{CD}$ and it cut them at G and F respectively.

Question:  Which is the transversal among the lines $\overleftrightarrow{MN}$, $\overleftrightarrow{IJ}$ and$\overleftrightarrow{KL}$?
Question:  Is there any problem in removing the condition ”different points” from the definition of transversal?

More Definitions:
Follow the previous two images. The angles 1,2,7,8 are called Exterior angles. The name itself suggest why so.
The angles 3,4,5,6 are called Interior angles.
The angles (1,3);(2,4);(5,7);(6,8) are pair wise Corresponding angles.
Angles(5,4);(3,6) are alternate interior angles.
And (1,8);(2,7) are Alternate Exterior angles.
Also, (5,3) and (6,4) will be called Consecutive Interior angles. They are the
interior angles on the same side of the transversal.

Now if there is two parallel lines and a transversal intersect them, will we get  some relation between the special named angles. Let us see.

Axiom 3(Corresponding angle axiom):
If two parallel lines are intersected by a transversal, then the each pair of corresponding angles evolved, are equal.

Here AB is parellel to CD and EF intersects them. Then according to the
axiom 3,

\[\begin{array}{l}
\angle A H E=\angle C G H \\
\angle A H G=\angle C G E \\
\angle E H B=\angle H G D \\
\angle G H B=\angle E G D
\end{array}\]

Axiom 4:
If a transversal intersects two lines such that a pair of corresponding angle is  equal, then the two lines are parallel.

Theorem:
If a transversal intersects two parallel lines,then each pair of alternate interior angle is equal.
Proof:
Let AB and CD are two parallel lines and EF is the transversal intersecting them at H and G respectively.

We have to show that,
\[
\angle A H G=\angle H G D \hspace{10mm}(1)
\]
and
\[
\angle B H G=\angle H G C\hspace{10mm}(2)
\]
$\mathrm{Now}, \angle A H G$ angle $\angle F H B$ are vertically opposite angles. Therefore, by axiom 1
\[
\angle A H G=\angle F H B\hspace{10mm}(3)
\]
Also by axiom 3
\[
\angle A H G=\angle F H B\hspace{10mm}(4)
\]
Therefore from equations (3) and (4) we have, $\angle A H G=\angle H G D$
Similarly we can show that $\angle B H G=\angle H G C$ (Q.E.D.)

Theorem(Converse):
If a transversal intersects two lines such that a pair of alternate angles is equal, then the two lines are parallel.
Theorem:
If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary. The converse is also true.

Lines Parallel to the same line

Let AB is parallel to CD and CD is parallel to EF . Then, does it mean that AB is parallel to EF ? The next theorem answers this question.

Theorem:
Lines which are parallel to the same line are parallel to each other.
Proof:
Let $AB\| CD$ and $CD\| EF$.
We need to prove,$AB\| CD$.

Now, by axiom3, $\angle H I B=\angle I J D$ and $\angle I J D=\angle J K F .$ Therefore,
\[
\angle H I B=\angle J K F \hspace{10mm} (5)
\]
Hence, by axiom4, $A B \| C D$. (Q.E.D.)