##### 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.)